If a living quarter is built in a basement, is 5/8 drywall required on that ceiling?
5/8" drywall is rarely required in single-family residences. It was used historically because ceiling joists are often 24" on center, and 5/8 resists sag. 

You can certainly use no-sag 1/2" if you like. If your joists are 16" on center, standard half inch may be fine, by I don't know why you'd risk it for the minimal upcharge. 

A fire-rated door is also probably not required. If this is a rental situation, check with your local building authority on both counts.