I too have heard not to pick up bunnies a lot, and I totally ignore it! They fear being picked up because they are prey animals, and think they are being carried off. My two bunnies are just under 5 months old, and I picked them up every chance I got, and now the male loves it, and his ears constantly stay high meaning he is comfortable. He also runs up to his door when I walk by or go over to see him. When I open the top of his cage, he stands up on his back feet for me to pick him up. I can't ever tell if the female likes it because she is a lion lop so her ears flop to the sides, and occasionally go back a little but not often.
Sometimes, the laying down in the cage can mean they don't want you bothering them. If they go flat down and put their ears flat on their back, they don't want to be touched and it's best to leave them alone. My male bun does this sometimes and other times he is fine.
The pooping is completely normal. Some people can train their bunnies to never do it, but it is something that just happens. If you want, keep picking up the poop and place it in the litter, sometimes this helps. They generally don't know they even dropped one.
I do wish my bunnies were as playful as yours! Chasing the kids and circling your eat! How did you get them to be that way?!