Meaning of "Hunky Dunky"

I located a transcript of the episode and found a little bit of additional surrounding conversation:

Penny: Okay, if you’re going to drink on this date just promise me you won’t overdo it.

Raj: Overdo what? Happiness? Freedom? This warm glow inside of me that promises everything is going to be all hunky donkey?

So the background is two characters discussing consumption of alcohol to excess. Penny advocates avoiding excess, while Raj argues he should allow himself to succumb to it. His statements are meant to indicate he desires the sensations that drunkenness will bring, so this context should hint that "honky donkey" is something positive that results from inebriation.

Further, the line is also spoken by Raj, a character from India who often mixes American idioms. In this case, he seems to have jumbled the common expression hunky-dory, described by Free Dictionary as:

adj. Slang Perfectly satisfactory; fine.

Putting this together, we can conclude that Raj is saying he wishes to obtain the satisfying sensation that excess alcohol might bring.

Jonathon Green, Chambers Slang Dictionary (2008) has this very brief entry for hunky-dunky:

hunky-dunky adj. {var. on HUNKY-DORY adj.} {1950s+} (US) fine, excellent.

It seems worth observing that Green identifies five other hunky terms of varying ages in the same dictionary: hunky chunk (mid-19C U.S., "to steal food"), hunky-doke (1940s U.S., "in good/proper order; functioning as required"), hunky-doodle (1900s U.S., "fine, satisfactory"), hunky-dory (mid-19C+ U.S. "wonderful, Excellent, first-rate"), and hunky-peroodlum (1900s, "very attractive and sexually inviting").

Like Green, J.E. Lighter, The Random House Historical Dictionary of American Slang, volume 2 (1997) equates hunky-dunky with hunky-dory, but then lists two early occurrences of the term:

hunky-dunky adj. HUNKY-DORY. [Earliest citations:] 1955 Ruppelt Report on UFOs 119: Our vast files of reports are in tip-top shape; and in general things are hunky-dunky. 1981 Louisville, Ky., man, age ca50: Now everything looks hunky-dunky.

A Google Books search turns up this item from Brown v. Republic Productions, Inc. (September 14, 1945), in Reports of Cases Determined in the Supreme Court of the State of California (1946) [combined snippets]:

SCHAUER, J. — The controlling issue in this case is basically the same as that of Brown vs. Republic Productions, Inc., ante 867 {161 P.2d 796} [1945], our opinion in which has been this day filed and reference to which is suggested for a statement of pertinent facts. The cases may be said to differ only in that the findings established that the extent to which portions of the original compositions were incorporated in the revised productions is substantially smaller here. The musical compositions involved are five: "All This and Heaven Too," "Peek-a-Boo," "I Could Love You Any Time At All," "Hunky-Dunky-Dory," and "Bonita Lolita."

And even earlier is Samuel Adams, The Gorgeous Hussy (1934) [relevant text not visible in snippet window]:

"All hunky-dunky-dory-o. She'd like to see you."

"I'll bet she would!" snickered Tony, with an obscene leer. Roderick Dow, Esquire, straightened his spare form. His face was dark.

"Did I ever tell you about marrying a millionaire-lady, Eb?"

These very early instances suggest that hunky-dunky originated as a nonsense extension of hunky-dory (as "hunky-dunky-dory") that subsequently (but not often) got truncated to just hunky-dunky. The earliest match of all, however, may be a nonsense name on the model of Humpty-Dumpty. From T.P.'s and Cassell's Weekly, volume 2 (1924) [combined snippets]:

Hunky-dunky had no sense. Bought a fiddle for eighteen-pence, And all the tunes that he could play, Was "Sally, get out of the donkey's way!"

This rope-skipping jingle appears in the company of a number of Mother Goose–style rhymes along the lines of "Charlie Charlie sole the barley/ Out of the baker's shop..." Subsequent versions of the fiddle-buyer rhyme feature different names ("Dancing Dolly" [1977], "Charlie Chaplin" [1982], "When I was young" [1990], and "Pooka Sullivan")—and different songs ("Ta-Ra-Ra-Boom-De-Ay" and "Over the Hills and Far Away"). In fact, the original personage in the rope-skipping rhyme seems to have been either Polly Perkins, who appears in "A Village Sovereign," in The Living Age (September 11, 1897), or Dancing Dolly, who appears in the version recorded in Alice Gomme, The Traditional Games of England, Scotland, and Ireland, volume 2 (1898). "Hunky-dunky" doesn't appear in any other version of this rhyme that I've seen.

From the Online Etymological Dictionary:

hunky-dory (adj.) 1866, American English (popularized c. 1870 by a Christy Minstrel song), perhaps an elaboration of hunkey "all right, satisfactory" (1861), from hunk "in a safe position" (1847) New York City slang used in street games, from Dutch honk "post, station, home," in children's play, "base, goal," from Middle Dutch honc "place of refuge, hiding place." A theory from 1876, however, traces it to Honcho dori, said to be a street in Yokohama, Japan, where sailors went for diversions of the sort sailors enjoy.

I take it to connote a feeling of optimism and success; but it also has an outmoded and provincial tone.