Yes to the former, no to the latter.
To clarify - The former is a possibility although the divorce papers would recognise the putative nature of the marriage as the reason for the divorce. They don’t.
“Can you be married but not legally married?
No. Marriage is a legal arrangement under civil law. That means marriage in the United States is restricted and governed by state laws. Therefore, in order to be considered legally married the parties must meet all the requirements for a valid marriage in the state where they will be married.”
http://www.answers.com/Q/Can_you_be_married_but_not_legally_married