Do women in the US get an updated Birth Certificate with their new name after they get married and change their name?

No. But the new name is often on the marriage license/certificate

No - you are not issued a new birth certificate.

No they do not. They can and should apply for an updated Social Security Card though reflecting the name change.

Although you will not receive an updated birth certificate after marriage, it will be important to keep a paper trail to show your new name after marriage,. Your marriage certificate is the link between your "old" and "new" names. It would be wise to apply for an amended passport and a new driver's license to show your new name.