I have already had my daughter and have received my disability check from the state of California. I took the standard 6 weeks, as requested by my doctor, but have only been paid for 5 of them. It seems that it is standard to not pay for the first week, but will I ever become reimbursed? If it is required, and you take it, can we really not be paid for it?

I don't know CA law specifically but in my state and many others, you would never get reimbursed for the waiting week unless your employer chose to do so or you applied vacation or sick time to that week. It is perfectly legal and quite common for the first week of a disability leave to be unpaid.