Q:

Is “what’s” a correct short form of “what does”?

E.g. “What’s he think?”
Usually ’s is short for “is” so I don’t know.

A:

The NOAD reports that 's is the informal contraction for:

is: "it's snowing"
has: "he's gone"
us: "let's do it"
does: "what's she do?"

A:

It is certainly used in that way. 
I don't regard the word "correct" as meaningful in this kind of question, so the answer to your question depends on what standard you choose to designate by "correct". 

A:

This can be a contraction of "what does", but I'd generally consider it a "colloquial contraction" insofar as it's a bit of a colloquialism to say "what's" instead of "what does".  Basically, if you're describing someone who's said "what's" to mean "what does", or emulating their style of speech, then it's OK.  Otherwise, it's a rather quaint contraction and I'd recommend against it.

