The best way to answer your question is to refer to your employee handbook or speak to HR directly.
FMLA is just job protection for 12 weeks.
In CA you get 6 weeks for a vaginal delivery and 8 weeks for cesarean and it is paid at a portion of your salary (I didn't collect this so I don't know the %). You can download the paperwork online for your dr. to fill out. This is pretty routine for them, though some offices do charge a fee (mine charges $30 per form). This is paid through short term disability and has nothing to do with paid family leave or FMLA.