Most employers in the US don't pay for maternity leave--I work for a VERY family friendly company and they don't even pay maternity leave for local employees. If benefits are available, they come from some sort of disability insurance. If you are paying for short-term disability, then assuming you meet the requirements of the policy, you can use it for income. You may or may not have any flexibility in how you receive the benefits.
Our policy picks up after 5 consecutive sick days. Then short term disability pays 100% of your salary for 26 weeks. The length of time you can draw from the policy all depends on the reason for taking it and when you are released for work from a doctor. Our policy pays for 6 weeks after a vaginal birth and 8 weeks for a c-section. I had a vaginal birth and get paid for sick days, so I ended up with 7 weeks paid and then used some vacation and unpaid leave to stay home 10 weeks.
I am not sure who you are referring to when you say "they also mandate that I take off 9 weeks". 6-8 weeks is standard post-partum in this country and is NOT mandated by FMLA--it is usually defined by the maternity leave or disability insurance. Of course, this can be extended if you experience complications,
You can actually exhaust all 12 weeks of your FMLA leave after the birth of a child...you just won't necessarily get paid for it. There is a "reasonable notice" clause that needs to be observed in order to utilize it all, but it is certainly available.
FMLA is federally mandated for all companies with a certain number of employees...it ISN'T a benefit. Please don't let anyone act like they're doing you a favor. You have obligations and responsbilities as an employee, but you have a right to that leave.
You definitely need to read your benefits summary and get with someone in HR. There can be a alot of paperwork that goes along with the disability insurance and doctor's offices aren't necessarily that helpful with getting it done.