Yes, they do. Believe me, you are not the first to have an early flight out, and you won't be the last. As was stated, the check-in desk is only open 4 am-1 pm, so your son will not be able to utilize it
The 16 year olds father would most likely not be able to accompany him. You'll get a transportation notice in your room the night before your flight, and if you don't have a boarding pass for that bus, you can't board. Your 16 year old will be fine going by himself. I'm a little bit older than him and have now gone by myself twice, so I can honestly say he will be fine. Just make sure he has some sort of government-issued photo ID. You could always ask, but I very much doubt it.
If possible, you might want to bump his flight back a little later (like 8 am or later) so that he can utlize resort check-in. That way, he wouldn't have to check in at the airport. You could help him check in at your resort, and then he can go to the airport and go straight through security.