Hi everyone! The whole benefits/leave thing is very confusing. I went through this all about a year ago, so I am only speaking to what my experience was. I also work in HR so I had a little extra help on my side;)
In CA, you are eligible for 4 weeks of leave PRIOR to the birth of your child. All your doctor has to do is write a letter and you turn that into edd to begin your claim. You are not able to tack that onto the 6 weeks you get AFTER the birth of your child (8) weeks if you have a C-section.
Once your 6 weeks of short term disability runs out, if you do not qualify for FMLA (which I did not) you can then apply for PFL (Paid Family Leave) and still receive the 55% of your pay. The state will send you the paperwork to fill out for that after the STD (short term disability) runs out. The stipulation with PFL is that it is and UNPROTECTED leave. You will want to talk with your manager about the leave and make sure that you will still have a job when you come back.
Here is a link I found with more info on PFL...
Another cool thing about PFL is that both the mother and father can take it. You have to use the benefit within the first year of the child's life and it is for bonding:) SO if mom has to go back to work, dad can take up to 6 weeks (55% of pay) to stay with the new baby. Also, you don't have to take all 6 weeks all at once. You can take a week here and there, however you choose.
FMLA as I understand it lasts for up to 6 months. I don't believe that you are paid the entire time, so I would google that for more information.
Hopefully this info was helpful :)