1. I think I heard this on Kiraz Mevsimi: Q: Napiyorsun? A: Hiç

I wanted to make sure I understood the answer. I thought it should be hiçbir şey. Is that only OK in conversation to answer with only hiç? Can other questions be answered with Hiç?

Hiç have same meaning, but it´s shorter.

2. If a child is not behaving well can I say başma? i.e. Don´t start.

Don´t start = Başlatma

Also you can say:

-Yaramaz olma

-Yaramazlık yapma

-Uslu dur

3. Is there a way to say "I will" on it´s own. For example "Say hello to so-and-so". Response: "I will". Is it OK in these siutations to just say yapacağım?

No. You can use tamam or peki for such things.

Example: 

Ahmet: Kardeşine selam söyle.

Burak: Tamam, söylerim.