Yes it's possible. They can can also get erections while sleeping

I know a man can have an erection during a dream, but what sounds so unrealistic is the statement that this man was relaxed enough to fall asleep and still excited enough to keep an erection. I would have thought the moment he fell asleep he would have been completelly relaxed until getting stimulated by a dream later on.

It is possible for a guy to fall/be falling asleep and still have an erection. It has happened on a few occasions with me and my boyfriend as he is currently chronically tired. It's not that he isn't enjoying himself, it's just that if he is on the bottom, is comfy, having a nice stimulation then its easy to fall asleep if you are/were already struggling to stay awake. I try not to take it personally and i know he doesn't want me to think it's my fault. Could be a similar thing with the OP, their boyfriend could have been extremely tired or have health issues and the action of being able to relax and enjoy themselves led to them falling asleep but still having an erection due to the stimulation/they are still enjoying it, just in a different state of consciousness XD Don't feel bad OP :) but maybe talk to your boyfriend about it if it does bother you, he would probably feel embarrassed about it as well.

Yes, but the pelvic thrusting would probably stop. And I'm not sure he could ****** and stay asleep.