Arya used to bite me A LOT! I just had to put up with having bitten fingers every day for a few months until she got used to it. She was a screamer as well, we used to joke that she was just a grumpy teenager as she screamed whenever I woke her up haha.
The best way to get him out of the cage though is to take his cage into the bath, open it all up and just let him climb out. He won’t be able to get out of the bath and you can then clean his cage. Once his in the bath get a toilet roll and get him to climb into it and then pick him up that way, rub your hands in his bedding so they smell like him and then let him run through your hands. He is probably going to bite you, but when he does DON’T put him back, he can’t learn that bites mean he goes back. Keep holding him and put him back after a while. Do it every single day and he will get used to it. A hamster bite isn’t really hat bad, they sting and just make you jump more than anything so don’t be scared of them!!!
It takes time and patients to tame some babies xxx
A tip: it sounds like your hammy has cage rage. Basically when they have a small amount of space, they get crazy defensive. Solution, buy them a bigger cage, and you should have a much happier ham.