Mama should have paid for Vint and Naomi's trailer to be fixed so they could have saved their money for their spot at the trailer park so they could finally be out of her hair. She should have taken action to do so.

Thelma might not have had all that much money. Probably some pension from Carl, but how much more? She didn't work outside the home (well, consistently anyway), and although the house was probably long since paid off I just can't envision her having a ton of $ to her name. Vint and Naomi certainly should have given that they had been living in her basement for seven years.
I cannot remember at the moment if the storyline had Vint and Naomi paying rent.

Vint did pay rent when he first moved in.

Vint did pay rent when he first moved in.
It was stated several times that he paid rent but it was also stated by Mama that he was either always late or he didn't pay.

Did Fran pay rent too? I always figured she did.

Did Fran pay rent too? I always figured she did.
Yes, she did. Remember her saying to Thelma, I think on the first episode when Vint and the kids are moving in and she is threatening to leave, that "she stands a very good chance of losing a paying resident(or customer, or something to that effect, I forgot the exact word she uses in reference to herself), if she lets Vint move in.