No

It's more difficult to do it in US since you were born and got married in China.

没有出生证 is not a problem. I believe most Chinese don't have birth certificate. You can get notarized birth certificate in China without it.

Ask your parents to go to city's notary for the procedure. It should not be difficult. I believe notarization will cost few hundrend RMB. But it's worth it as you will need them from time to time.