Well, in Cantonese, "ching" (pronounced more or less tsing or tseng) and in Mandarin, "qing" (not too different in pronunciation) these words mean "please", and it is often used in the sense of inviting someone to do something, like "ching yam" would be please drink, "ching yap leih" is please come in, etc.
Now, usually people drop the verb and just say "ching" to mean "please drink" or "please follow me" etc.
Furthermore, in Cantonese they double up many words for certain expressive effects, so that might have contributed to the double "ching", or "chin" as well.

And compare 'chop chop' (hurry up), from Chinese pidgin English (the OED now gives k'wâi-k'wâi, but I'm sure I read somewhere else tsiap-tsiap).

You can search for "qingqing" with:
Traditional:
http://www.google.com/search?q=%E8%AB%8B%E8%AB%8B&ie=utf-8&oe=utf-8
Simplified:
http://www.google.com/search?q=%E8%AF%B7%E8%AF%B7&ie=utf-8&oe=utf-8

Mandarin "kuai" corresponds to "faai3" in Cantonese, so no idea where "chop" comes from.

(Y.) should mean one of the authors of Hobson Jobson, Col. Henry Yule.

Yes, I know it stands for Yule, but I presume when it's used in the context of a citation like that, it means the citation is from Hobson-Jobson.

Answers.com on "chopstick" says:

Pidgin English chop, quick (probably from Cantonese kap; akin to Mandarin jí)

http://hpcgi2.nifty.com/Gat_Tin/fang.cgi?guoyu=ji2 gives results including:

急 キュウ ji2 jiek4 kip4 gib4 kip6 gap1

whose character seems mostly likely to match the meaning.

Hobson-Jobson is quite wrong in taking the history of the expression "chin-chin" back to William of Rubruck, as the "chin-chin" in his story is a completely different word to the later "chin-chin" that derives from Chinese qing qing 請請. William of Rubruck's "chin-chin" (or "zinzin" according to Francesco Carletti) is a representation of the Chinese word xing xing 猩猩 (a type of ape, but in modern Chinese the word for orang-utan). See here for further details.

Ng, as in yong jiu da xingxing 用酒打猩猩.

caffeind, I am a bit skeptical about 急急 (Cantonese geb geb in the pinyin-based transcription) a the sole origin of "chop chop". There is the rhyme and the rhythm, and of course jiji appears in many expressions (my personal favourite is the bureaucratic-exorcistic jiji ru lüling 急急如律令) but the initial consonant seems too different to my badly informed ears. I don't know anything about the formation of Pidgin, but would a mixed origin taking in account both the meaning and onomatopeic value of the English verb "to chop" be plausible? As in

Ayo swing swing swing, to chop chop chop
Yo that's the sound when MCs get mopped
Don't come around town without the hip in your hop

(A Tribe Called Quest, "Keep It Rollin'", Midnight Marauders).

This is far from satisfying, but I remember you linking to Le Monde's editors' blog (something about "vavavoom", if I recall correctly), and they happen to have posted this two years ago: Tchin-qing!

Nothing new, just a quote from Pimpaneau explaining qing... [gesture]! qing! 請... 請...