For an unborn child, which parent has custody? Also, if the father does not sign the birth certificate in California, does he lose his paternal rights? And can child support be collected?

There is no custody for a unborn child, since the child is not born.
You will have to wait till the child is born to file for custody. No the father does not lose any rights by not signing the birth certificate, he will have to prove by DNA latter if he files for custody.
And yes you will have to prove by DNA that he is the father if he is not on the birth certificate but you merely get a court order for the DNA test, he is proven to be the father then you get the child suport.