I exercise if I can't sleep. Which is almost every night lol.
I start by going for a run around the neighborhood. If that's not enough I do sit ups or lift weights till I finally fall asleep. For getting back to sleep? More of the same exercises. It works for me. Just work my body till I can't go any longer and just fall into my bed and wake up X hours later.
As for early morning games? I like JRPG's, something about waking up and staring down a hundred hour long monster feels great lol. Right now i'm working on replaying Dragon Quest 9 and Disgaea D2. ...that ones going to take a lot of early mornings.
As many others have said, podcasts have been an absolute lifesaver for me. I'm a pretty anxious person and my mind tends to wander if I don't keep it occupied. A chilled out podcast puts me to sleep every time, to the point now where I'd probably struggle to go to sleep without one on in the background! No wonder I'm single.
I have pretty bad insomnia on and off. I usually fall asleep listening to podcasts. If I wake up and can’t fall back to sleep lately I’ve been watching old Game Room quick looks. Those usually make me pretty sleepy. Or I read. A night shift filter and blue light filtering glasses have seemed to also help with the phone not contributing to the sleep problems.