1. Do I really need to provide more than a certified copy of our City of New York marriage certificate as marriage evidence, e.g. an acknowledgement of deposit of a copy of the certificate with the Home Office? No, nothing other than the copy of your official marriage certificate is necessary.
2. How much evidence of my husband's and my relationship do I need to provide if we have a son with a UK passport? Are joint US tax returns and joint US bank account statements necessary/sufficient? You will still need to provide proof that your relationship is genuine and ongoing, so chats/texts, photos, proof of current cohabitation, etc from the start of your relationship to the period before your application is submitted. I wouldn't send any bank statements other than those that you are using to prove that you meet the MIR, to minimize any possibly confusion your ECO may have about what is relationship vs what is financial evidence. If you have been living together then you could show proof of that through utility bills, lease, mortgage, etc. I know it seems silly when you have a child but UKVI assumes nothing, so you have the burden of proof