Not really answering your questions, but this thread is timely for me. My twelve-year-old is working on pushups and chins right now.
She's still tiny (5'3" 98 lbs, all knees and elbows) so the BW moves are progressing really well. She likes to try to out plank me.
The inverted rows (fat man pullups) are a fantastic idea for getting her to activate, and start building her back and arms. I do them all the time myself, and I don't know why I hadn't started her on them. Thanks for the reminder, More_Squat.
We don't have a BB setup at home, and my gym only allows kids under 18 on the weekends, and not in the free weights. In that situation, goblet squats with a KB or DB are good. Also, she can farmer carry some DBs just to work on grip strength and conditioning.
I've been doing some complexes with just the bar, and I think those might be good for kids, as long as they aren't hurrying it up like it's a race. Maybe with something lighter like those weighted sticks that are 15 to 20 pounds or so, adjust for your kids' strength levels. I haven't tried this with her, but something like 8 of each of these, then rest a few minutes and repeat. RDL, Bentover BB Row, Power Clean, Push Press, Back Squat.
For my daughter, mixing it up with running drills, playing catch, or badminton helps hold her interest. I think as much as it can feel like play for them, the better. On Saturdays, she'll use some of the cable machines at the gym.