Sure they can. Use any other email address at domain.com to register.
Yes. There are solutions to all of these issues, but what often happens is these situations come about through the natural course of companies changing over time - different people managing accounts, different providers, etc. The happy path is easy, but the happy path is rarely the one we find ourselves walking down when we inherit a previously made decision.
1. Use "[email protected]"
2. Let the domain registration lapse
3. Someone else registers the domain and now can't create an AWS account.
Rare but not impossible.