Fonzie is acting out hormonal mating behavior. He is looking for a cavity-like corner where he can build his nest. Put pieces of wooden toys in his cage so he can chew on them there, instead of going after your walls and daughter's things.
You are right. If the house is NOT bird-proofed, then allowing him to walk around is quite dangerous. I discourage my birds from climbing off the cage by putting "skirt-barriers" around the cage. It works like a charm with my Sweetpea who will NOT go over the line. She used to climb down to run over to try to attack Merlin, but no more.
Specifically, I got plain material from a sewing store... and velcro. I cut the material to the parameter of the cage (or you can have 4 separate pieces for each side of the cage). Four pieces of velcro were cut and placed on each of the four sides of the cage. The material was wrapped around the cage legs and held up by being attached to the velcro. The material wrapped around the bottom of the cage has posed as a barrier that has stopped my birds from climbing down. If Fonzie does not try to jump or fly off the cage, this should help for a time.
Also, you can use some positive reinforcement techniques to "REWARD" him for staying on the cage. Give him LOTS of praise when he behaves and stays on the cage... and a food reward would help, too. Find a treat that he LOVES and stop serving in cage. Instead, give from your hands. Break it into bits so you can serve more often, like cashew bits or one sunflower at a time. Do NOT feed too many. Anyway, use that as positive reinforcement to reward him for staying on the cage.
Hoping these ideas help. Will keep thinking about this for you.
Blessings,
Maggie-