Do you need to file a state return, if you have a Zero balance?

If by "Zero balance", you mean you don't have to pay a tax with your return, you should still file the state return. 

The only reason you know that no tax is owed its because you prepared the return. If you don't file the state return, the state won't have all the information you used to prepare the return. They probably know a lot about your income, but not necessarily about all the deductions, exemption and credits you might have.

Yes

It depends.  Each state has its own filing requirements.  More information about your state may help resolve your question.

@huggirl67