I think the baby name is Shaun by default and the different names might be for the player (like Mr.Howard in the E3 demo). Correct me if im wrong.
No, there is 1000 player character names recorded. So if you choose one for your character, that is part of said recorded names, Codsworth will say your characters name.
No, 1000 different recorded names for your own character. As far as we know, the baby is always a boy and always Shaun.
Chekov's Gun would indicate we will see the baby in some meaningful fashion in the 'third act'. It would be tough to accommodate gender and names into that dialogue.
Yeah, my first character was actually going to be named Sean. Unlucky coincidence, haha.
Mistake
Oh I thought this was what we are going to name our in game child hah.
Well, we can assume that every action that occurred with the male protagonist will be reversed if you play the female protagonist. Your husband will be sitting on the couch when you answer the door, your husband will tell you to check on the baby and your husband will be the one carrying the baby when you run for the vault. It could be likely that the baby's six will change depending on which gender you choose to play and the husband will say "how are my two favorite gals?" Of course if names are set it's probably pretty likely that the girl will be named Shawna.