what's he got/what is wrong

The first one is very informal and shouldn't be used in writing or in formal settings. Your young son wants to play with a friend, but the friend's mother says he's home sick today. Your son might say that. If there was some reason you would need to say this in a formal setting, you might say "What does he suffer from?" or somethign like that. I can't really think of too many times you might need

What's he got wrong with him?
= What is wrong with him?
= What's the matter with him?
= What physical or mental disease, disability, malfunction, or impairment, or symptoms thereof does he have or suffer from?

These expressions can be used figuratively.

-- My car won't start.
-- What's wrong with it?

-- I was standing in line waiting to bu

What's he got wrong with him?
= What has he got wrong with him?
= What has he wrong with him?
= What does he have wrong with him? (all incorrect)

Cf.
What's he got?
= What has he got?
= What has he?
= What does he have?

I think it's all right to say: Nothing is wrong with me.
I don't think it's correct to say: Nothing has wrong

As Grammar Geek has already stated, the 'have got wrong' form is very informal.

You are quite correct to point out that what's is what has in the context above.

has got is an idiomatic form of has. That's also true.

What does he have wrong with him? does not strike me an incorrect, but as a lesser used, somewhat wordy variant of