Can I collect back payments of spousal support?

In order, the answers to your questions are "maybe" and "yes." Generally, a support order creates a "judgment" as to each payment which comes due and remains unpaid, and generally the 6-year statute of limitations runs from the date of each such payment coming due. But there are a lot of details, starting with the order in question, and including the possibility of "renewal" of judgments. Consider a consultation with (or at least a phone call to) a family law specialist well versed in collections law to go over the specifics of your situation.