Hi and welcome to Binky Bunny!
When rabbits are first brought home, it’s usually a good idea to limit their space for the first few days or first week. They need a little time to settle in before they can start exploring the house. And you need to make sure that any rooms they are loose in are bunny proofed. They of course need out of cage time for play and exercise. It would have been a good idea to attach an x-pen to the cage, then you could sit beside it or in it with him. Some rabbits are very cautious and it may take weeks or months for him to feel really comfortable with you and in your home. Rabbits do like hidey places – under and behind furniture. Treats can be a good way to get him used to you, but you don’t won’t to overdo them. Just sitting quietly near him, even ignoring him, may make him want to come and investigate you or interact with you.
If he has been peeing, he must have been drinking water. If you have started him on any greens or veggies (go slow with that if he isn’t used to them) you can make sure that you sprinkle a little water on them before you give them to him
The vet said that he had been neutered, and if he hadn’t, that would be pretty apparent since he is almost 2 years old. Once a male rabbit reaches about 4 months of age or so, you will see his testicles. The reason I mentioned him being neutered is that it isn’t always easy to litter train a rabbit that hasn’t been spayed or neutered. Was that the shelter vet who told you he was neutered or your personal vet? If the shelter neutered him, it should be in their records. Sometimes a rabbit can kind of suck up their testicles and you can’t see them, but when your rabbit is more comfortable with you, you can check to make doubly sure. You can even hold a treat over his head to make him stand up, and you would be able to see if he has testicles or not.
To get a rabbit back in its cage, give it a treat that it only gets at that time, like a small slice of apple. Or,if it’s bedtime, give him his pellets then, and only then. Hopefully he will want to go in for those.
Rabbits love to dig, and a litter box is one place that they do it. You can buy litter box screens to keep the contents in. You could provide him with a diggy box outside of his cage, filled with shredded newspaper, and maybe he would dig there instead. Some rabbits aren’t that fond of toys, or they lose interest in them quickly, so it’s not too unusual if he doesn’t play with his.
Picking up rabbits can be tricky, since they don’t like it and it scares them. You have to start slowly with that. If you are siting beside him, pick him up for a few seconds, then put him down. You could pat him afterwards or give him a tiny treat. Keep trying that until he seems comfortable enough for you to pick him up for more than a few seconds. Hopefully he will become calm enough at some point to allow it, for you to pick him up to put him back in a cage or to put him in a carrier if he needed a vet visit. My own rabbits hate being picked up, even though I have had them for a while. That’s a struggle for a lot of people, and we have to be careful that the rabbit doesn’t struggle and flail around and injure themselves.
It would be great if your rabbit could be totally free range, but it might be a little early for that. Try to keep him in a smaller area – cage + x-pen, then slowly expand his range.