"What is this?"
Those are different case-forms.
Что is nominative (or accusative) case, it’s used when this word is a subject or direct object.
Чем is instrumental case, it’s more like ‘using what? with what?’.
Чего is genitive case, it’s more like ‘of what?’. There is a tendency for чего́ to replace что in colloquial speech (often in a shortened form чё), but this is markedly non-standard.
«Ха-ха» (or just «ха» for a small chuckle, or «ха-ха-ха» or even longer for longer laugh) is a normal, loud laughter.
«Хе-хе» a laughter you’re trying to hide (e.g. it may a laughter on a topic that is not really funny, maybe a laughter because of shadenfreude).
You hear correctly — что indeed sounds as if it were spelt што, and Wiktionary gives the IPA [ʂto].
I too want to find out under what circumstances ч should sound like ш. As there has been no answer here so far, I searched further on the Internet. According to http://www.russianforeveryone.com/Rufe/Lessons/Course1/Introduction/IntrUnit10/IntrUnit10.htm, this happens in a few words, and we’d just have to learn them one by one. Aside from что, other common instances are чтобы (in order to), конечно (of course, sure), and скучный (boring or bored).