I can relate. My military service and discharge is in one name, some of my transcripts are under Mrs. Robin Firstmarriage, and everything else is under Mrs. Robin Doors. Trying to get everything to match up is a treat, let me tell you.
Nope. They can’t. More than ten years ago laws were changed. If a parent is a parent via birth certificate (or whatever passes in the country of birth) or adoption paperwork, that child automatically gets U.S. citizenship. The immigration officer has no decisions to make. (And parents who adopted internationally had a lot fewer hoops to jump through).
Same thing for me. I had occasional issues with my own biracial kids with a different last name, but people always assumed I was my Caucasian nephew’s mother despite different last names.
There was some calling me " Mrs Kidslastname", but that really didn’t bother me.