How should I respond if they are separated?
That would depend on if your parents live together or not. If your parents are married but are considered legally separated by a state, or if they legally married but have chosen to live separate lives then they are considered separate, as if they weren't married.
What if my parents don't want to provide the required information for the FAFSA form, can I submit it without that info?
You could, but take into consideration that you would not receive an EFC and some state-school-based aid programs look at the EFC to determine your eligibility for their funds, if you don't have an EFC then you won't be eligible for those aid programs.