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"Lukewarm" isn't really much different than saying "tepidly warm", which is something that people definitely say. And it makes sense in the same way that "yellow orange" makes sense: "luke" was probably considered less warm than "warm", with "lukewarm" falling somewhere in the middle. (This is definitely the case for "tepid", which seems to be the closest synonym of "luke".) In any case it made enough sense that there's also the (dialect) synonym "lewwarm". Both "lewwarm" and "lew" (by itself) are still used in Scotland. If you refer back to Oxford Dictionaries, "luke" is thought to be derived from "lew", which is why this is relevant.

I haven't been able to find another semi-recent example of standalone "luke" except for the one in Dickens. Looking at the OED's citations, the others are all from pre 1500, with two examples occurring after that ("luke-hearted", ?1507; "lukeness", 1597).

There are plenty of examples from Middle English of "luke" describing things other than water:

Opened wes his breoste; þa blod com forð luke.

"Opened was his breast; the blood came forth luke."
_{Laʒamon's Brut}

It could even describe people...in the exact same way as "lukewarm" can:

He is fyeble and lheuc to alle guodes to done.

"He is feeble and luke to all God has done"
_{Ayenbite of inwyt (transl. Michel Of Northgate)}

More examples can be found in the freely available MED.

The following sources appear to support your suggestion that lukewarm is etymologically a reduplication of warm:

From Word Detective:

The Oxford English Dictionary (OED) does indeed employ “tepid” in its definition of “lukewarm,” and, logically, lists “lukewarm” as a direct synonym in its definition of “tepid.” The two words are, in fact, nearly identical in meaning, with the only shade of difference in usage being that “tepid” is more often applied to liquids than to solids (“Let the Water stand in the Sun till it grow tepid,” 1691).

.... “lukewarm” represents a combination of “warm” with the somewhat older English adjective “luke” (or “lew”), which itself meant “warm” (meaning that “lukewarm” etymologically amounts to a redundant “warm-warm”). “Luke” came from the Old English word “hleowe,” which meant, amazingly, “warm,” and which in turn seemed to be rooted in an Indo-European root word that meant “weakly warm.”

From World Wide Words:

Lukewarm has been spelled in all sorts of different ways down the centuries, including lew-warm, loo-warm (a necessity in our house), lewke-warm and luckwarm. The first part was mainly in dialect use and transmitted orally, so the spelling only settled down to our modern version in the eighteenth century.

Luke has, of course, nothing to do with the given name. It comes from an Old English adjective hléow that has modern relatives in Dutch and German. It may be linked to hlēo, shelter or lee, and also to another Old English word meaning debilitated that developed into lew, weak or wan. To be lukewarm is to be only weakly warm, tepid.

An odd sidelight is that from the thirteenth century, luke by itself could mean lukewarm, as could lew (the English Dialect Dictionary reported a century ago that it was then very widely used in various spellings throughout England, Scotland and Ireland). So you could argue that lukewarm means “warm warm”.

From The Grammarist:

Lukewarm describes something that is tepid or only slightly warm, something that is neither hot nor cold. ........the word lukewarm is derived from the Old English word hléow which means sunny or warm, which evolved into the Middle English word lewk. The word luke was once used on its on own to mean warm, and this use survives in the word lukewarm.

As a side note, luke also meant “nothing” as suggested by Green’s Dictionary of Slang

luke n.
[? northern dial.]

nothing.

• 1821 [UK] D. Haggart Autobiog. 17: He quized his brother for having given us so much trouble about luke.
  

I would like to add something from my mother language, Dutch.

We can almost directly translate "lukewarm": "lauwwarm". But we do use "lauw" separately as well, like "Mijn thee is nu lauw." ("My tea now is lukewarm."). As we are lazy speakers, we often leave out "warm".

We can use it figuratively, like "Hij reageerde nogal lauw." ("He reacted quite 'luke'.), in which it means something like "not very enthusiastic", in the same way English-speaking people can talk about a "warm welcome" but now it would be a not-that-warm welcome.

According to a Dutch etymology website which cites lots of etymological dictionaries, "lauw" is related to the English "lew". It comes from the Latin "caleo" which means warm. This same site talks about "warm" coming from Greek "thermos" and Latin "formus", meaning "heat".

To answer your question, I don't think "lukewarm" is a perfect example of reduplication, as "luke" and "warm" actually do not have exactly the same meaning. It is more like "yellow-green" when you mean something between yellow and green, although I do not really feel like "lauwwarm" is warmer than "lauw"... Interesting.

There indeed is a (West) Frisian / Dutch connection (I'm Frisian myself): a Dutch cognate is "leuk", which now means "fun, amusing, pleasant" and a Frisian cognate (in the 19th century still current, nowadays no longer) "lûk (adj.)" (not at all related to the verb "lûke" (pull) BTW) which meant "clever, cunning". The Dutch word used to mean, according to the WNT (the OED equivalent for Dutch) "calm, relaxed" (see the lemma in Dutch here) and quite relevantly "warm up" in "opleukeren" (a verb derived from that root); other such forms exist in Dutch and Low German dialects as well, all pointing to Germanic -u- in open syllable (so lengthened).

As can be seen in the linked WNT lemma it has been popular as a modifier in words as "doodleuk", "leukweg" (that still refer to the meaning of calm and composedness).

My Dutch etymological dictionaries find the connection to "lauw" (lukewarm/tepid) (which corresponds to the Scottish "lew" forms) unclear and somewhat implausible (vocalism doesn't match, and whence the k?), but maintains that "moderately warm" must have been the older meaning then shifting to just moderate, then calm, and then (in slang?) to "fun", like words as "cool" have done...

Maybe the extra "warm" after "luke" was meant as a clarification, when the sense of the word "luke" wasn't that clear anymore, e.g. like in Dutch a whale is called a "walvis", which was originally just a "wal", but that word form could also mean "wall", hence the clarification "vis" was added for the animal (I know, it's not a fish biologically..), so such formations for clarification do occur sometimes.

'Luke' in the compound 'lukewarm' functions as a downtoning semantic reduplication, supposing 'luke' is considered to have the (perhaps) more modern sense of "a moderate degree of warmth, tepid" rather than the (perhaps) earlier sense of "weak, little". In some other compounds containing 'luke', 'luke' downtones the second element. Examples of such downtoning, although thin on the ground, include 'luke-hearted' (and, possibly, 'Lukeheart') and 'luke hot'.

In English etymology, semantic reduplication is mentioned by, for example, Liberman (An Analytic Dictionary of English Etymology: An Introduction, 2007), with reference to 'ragamuffin' (emphasis mine):

The most convincing hypothesis traces both rag- and -muffin to words for 'devil,' as in OF Rogomant (though in French it may have been a borrowing from Germanic), preserved in E Ragman and Ragman's roll (> rigmarole), and Old Muffy, from AF maufé 'ugly; the Evil One.' Ragamuffin is then a semantic reduplication with an augment (-a-) in the middle, *'devil-a-devil.'

A 'downtoner' and so 'to downtone', in the linguistic sense, approximate the opposites of 'intensifier' and 'to intensify'. Both intensifiers and downtoners are qualifiers. Intensifiers used ironically sometimes function as downtoners; often, negating an intensifier converts it into a downtoner.

Hence, 'luke' in the sense of "tepid, moderately warm" downtones 'warm' in the compound 'lukewarm'.

For the etymological influence on 'luke' of Anglo-Saxon wlæc, the older meaning of which was 'weak', see Skeat 1882, An etymological dictionary of the English language (under the headword 'lukewarm', after the pilcrow). This influence suggests the sense of 'lukewarm' may derive from elements meaning "weak-warm", a meaning which accurately represents contemporary use.

For a suggestion that 'luke' was used dialectally in the sense of 'little' see Cockeram 1623, An English dictionarie, as cited by Robertson in Life on the upper Thames (middle of last paragraph, p 19, where "Cockerham" is either a spelling variant or an error). Dialectal use of 'luke' with the meaning "little, slight", when compounded with 'warm' into 'lukewarm', contributed to the contemporary sense of 'lukewarm': "slightly warm, a little warm".

As mentioned and exampled by Skeat 1882 (op. cit.) from Dickens, etc.,

Luke means 'tepid' and can correctly be used alone,

as well as to downtone or otherwise qualify various signifiers: warm, blood, bath, water, fire, hot, and heart.

Some Paywalled Attestations from the 20th Century

'Luke-hearted' survives primarily as a family or nickname. One example of the latter is this "Personals" ad from The Los Angeles Times, 18 May 1960:

PATRICIA Gentile, aka Lukeheart, call MA. 5-1494. Urgent.

Attestation of 'Lukeheart' as a family name is too common to be worth exampling. Other attestations of 'luke-hearted' and variants are vanishingly infrequent in the 20th century. They include a coinage in The Times Herald (Port Huron, Michigan) 31 Jul 1967:

Naturally, the two find ways to help each other in this luke-heartwarming episode [of the television show "Coronet Blue"].

Another instance is this from The Courier-Journal (Louisville, Kentucky)
20 Jul 1988, reprinted a week later in two other newspapers, then 9 years (1997) later in the original venue:

People used to refer to nice folks as luke-hearted, which means, of course, warm-hearted. But that word seems to have disappeared, which is too bad.

The author of that article seems to have misconstrued the usual sense of 'luke-hearted' in the 19th century, which was more akin to a much-weakened or neutral 'disheartened' than "warm-hearted".

Examples of 'luke-hot' from the 20th century are more various, as follows.

In the Daily News (New York, New York), 17 Feb 1946, the intended meaning of 'luke-hot' may diverge from the expectable sense of "slightly or weakly hot":

June introduced some luke-hot letters which indicated that Schreiner's ardor had been considerable.

The meaning is as expected in the Edmonton Journal (Edmonton, Alberta, Canada) 01 Apr 1964, but I hesitate to ascribe that accuracy to a hypothesized more-literate Canadian audience:

 Even if you have failed to see the Oilers play this season, you know they've been white-hot because the cliche-riddled news services have been calling them the white-hot Oil Kings since last October.

 When will the news people stop? As soon as the Kings demote themselves to grey-hot, or luke-hot.

From The Miami News (Miami, Florida) 15 May 1968:

 In fact, my independent survey showed these percentages of old-school, step-outside-and-say-that, rabid patisans in either party:

 McCarthy, two per cent. Kennedy, two per cent. Nixon, two per cent. Rockefeller, two per cent. Utterly baffled, 91 per cent. (The other one per cent is still for Goldwater.)

 This doesn't include the great numbers of luke-hot supporters, of course: The millions who stand around in the booth and eventually vote for somebody....

From The News Journal (Wilmington, Delaware) 23 Jan 1973:

 Delaware's last governor got into political hot water. If it happens again, the experience should be less painful, for the state is going to reduce the temperature of the water that comes out of its "hot" faucets.

 ....

 Would you believe a future administration getting into luke-hot water?

From The Sydney Morning Herald (Sydney, New South Wales, Australia) 12 Jun 1989 (reprinted with slight editing in 1993):

...James Woods, as the shady nightclub owner Jake Wise [in the movie Against All Odds], delivers a nice performance. But apart from a few luke-hot scenes between Ward and Bridges that gave the flick its R rating on the big screen....

From the Santa Maria Times (Santa Maria, California) 21 Jan 1991 (the "Luke-hot List" trope was repeated for several weeks in the column):

To add to this tangle, production controllers also invented six manual lists to override the formal systems: the Hot List (angry customers), the Luke-hot List (moderately angry customers), ....

In the 'luke-water' compound, 'luke' is used with the sense of "slightly or weakly warm". Following are six examples of the compound from the 20th and 21st century popular press.

Belle Plaine News (Belle Plaine, Kansas) 03 Nov 1910:

...wading and bathing in the luke water of the Gulf of Mexico.

Akron Evening Times (Akron, Ohio)
15 Apr 1920:

For every two gallons of warm (LUKE) water you may need to add one-half pint of the above mixture, ....

The Journal News (White Plains, New York)
03 Mar 1952:

...add a pinch of soda dissolved in luke water water....

McKinney Weekly Democrat-Gazette (Mckinney, Texas)
05 Jul 1956:

A good way to remove chocolate stain is to pour luke water from a height of about two feet through the stained area of material.

Janesville Daily Gazette (Janesville, Wisconsin)
09 Nov 1967:

The easiest way to remove the skunk odor is to wash the dog thoroughly in luke water with plenty of Lifebuoy soap.

Calgary Herald (Calgary, Alberta, Canada)
02 Dec 2001 (headline):

Mark Wahlberg in luke water.

Other Evidence

Attestation of the compounds 'luke-hearted', 'luke hot', 'luke water', and 'luke bath' appears in the 19th and earlier centuries. For an example of the last, this from the 1880 Textile colorist:

...with methyl-green finish dyeing in a luke bath.

Attestation of 'luke' qualifying 'blood', and 'fire', absent compounding, appear in the OED citation collection. They can be seen without going through a paywall in the earlier NED, under the headword 'luke'.

I don't think lukewarm is an example of reduplication. Warm and lukewarm are not synonyms. Something that is 'warm' is hotter than something that is 'lukewarm'.

A (nice) 'warm' bath is a comfortable place. A 'tepid' or 'lukewarm' bath is usually one I have stayed in too long and want to get out of... unless I add more hot water.

Onomastically, the word Luke is a Latin as well as an American baby boy name, which denotes light giving.

The word "lukewarm" itself seems to give the redundant meaning "warm warm", but it's actually not. Here the name "luke" radically [indirectly] has something to do with the Old English adverb “hlēowe”. Which accurately suggests the meaning of "lukewarm" as "safely or comfortably warm". See the following reference from Today I Found Out.com:

It turns out, while today using “luke” to mean “warm” has gone out of fashion, possibly due to the popularity of the name “Luke”, at one time that’s what the word meant. This came from the fact that “luke” derived from “lew” or “lewk” or “leuk”, in Middle English, which meant “tepid” (slightly warm). This in turn came from the Old English adverb “hlēowe”, which means “warm or sunny”. Finally, “hlēowe” came from the Proto- Germanic *hlēwaz, meaning “warm”.

To a question similar to yours, The Hindu.com says:

I understand that the word ‘luke’ comes from the Middle English ‘leuk’, which in turn was borrowed from the Dutch ‘leuk’ meaning ‘tepid’ or ‘weak’. Since tepid means ‘barely warm’ there was no reason to add ‘warm’ to ‘luke’ in ‘lukewarm’!

To address your direct questions:

From the title question, I think you've answered this already from etymonline. From the OED first dated citations, at the time 'lukewarm' was used, 'luke-' meant 'tepid' not warm. So it was not perceived as a repetition of meaning (pleonasm). Reduplication is a repetition of sounds in sequence.

To your enumerated explicit questions:

1. 
   

   OED gives a number of terms beginning 'lukewarm' (lukewarmness, lukewarmish, etc.) and gives a single other word

   
   
   

   
   

   lukecherry - The name of a variety of cherry.

   
   

   
   
   

   with no etymology, but three citations all capitalized, leading me to believe it is a person's name and not realted to 'lukewarm'.

   
   
   

   
   

   16.. MS. Ashm. 1461 lf. 19 The Luke ward Chery ripe June the 10.

   
   
   

   1664 J. Evelyn Kalendarium Hortense 68 in Sylva Cherries..Luke-ward, early Flanders [etc.].

   
   
   

   1707 J. Mortimer Whole Art Husbandry (1721) II. 297 The..Lukeward, one of the best of Cherries.

   
   

   
   
   

   Given that the OED is so comprehensive, the absence of evidence is very strong evidence of absence of any such other 'luke-' word. If there is one it must be extremely rare.

   
   


2. 
   

   Reduplication is a repetition of the same word or morpheme to get a slightly different meaning, either intensifying, plurality, or some other twist. So the answer to your second question is 'no'.

   
   
   

   Maybe you mean some sort of pleonasm (a repetition of meaning), that is, maybe 'luke-' is a word that means 'warm'. The etymology, as you pointed out, seems to support that weakly in that historically (a long long time ago) 'luke-' may have meant warm. But currently it has no such meaning or implication. Because of this lack of extant meaning, folk etymology would yield a meaning of 'weak' or 'light'.