That is a very good option, nowadays in the morning I either go for a walk or downright do a full workout with Bodbot, it's a pretty good app.
No, I just do as many pushups as feels good to me. Started at 15, then dropped to 10 the next day because I was sore. This was a wake up call, because I used to do three times as much 10 years ago without getting sore. I can see now how my lack of exercise may be a large part of being tired.
I like the “no gym required” approach to making exercise a habit. A few push-ups, some donkey kicks, some hydrants, and maybe some hip thrusts or toe touches. It’s easy to get going and builds core strength & stamina.