The question word is always followed by the verb. But the word order is flexible after the verb.

The word order is topic, focus, verb, the rest. (Not all elements are always present.) As bdori said, the question word is always followed by the verb. This is because question words are always in focus. The sentence basically means "As for Péter, who does he know?" If you put Kit first, it's like "As for who, it's Péter who knows them." Doesn't make a whole lot of sense ...

I think it would be Ki ismeri Pétert? or Pétert ki ismeri?

You need to make Péter accusative, so he gets a -t. And because we're talking about a definite Péter, we need the definite verb form, which I'm guessing we will learn later.

If you use ismer instead of ismeri, it would make it basically like "Who knows anyone called Péter?", I think.