Yes, he’s probably taking some time to get adjusted and to get used to the new sights/smells/noises. Also if his space is on linoloeum/hardwood that could be the problem. My bun never goes into the kitchen for that reason.
You mentioned that he peed on your carpet a couple times. First of all… for cleaning, use a solution of 50/50 water/white vinegar – it does wonders at getting up stains and smells. Getting rid of all the smell is important because if the carpet smells like his pee, he’ll start to associate those place with his pee spot. Some buns just like peeing on soft surfaces like comforters or the couch/bed, so you may need to restrict his access to certain locations until he is completely litter box trained. Coming from the shelter he may not have had the reinforcement he needs for litter box training. There are tips on this site for how to litter box train your bun. Also, he might be peeing to mark his new place – claiming it as his.
Is he neutered? Neutering him will cut down on marking, improve litter box habits, and reduce aggression.
You’ll want to slowly expand his space, especially if he’s having trouble with his litter box training. So much space at once might be overwhelming, so start by blocking of a portion of the room with a fence or gate, or just pile up boxes, bags, chairs etc to block off part of the room, and then gradually make his space larger. Is his cage a cat carrier? Or is it bigger than that? If you keep him caged for any length of time, he’ll need a cage or a pen much larger than that – make sure that when he’s in his cage he has access to unlimited hay, fresh water, a litter box, and a few toys. If you’re planning to eventually let him free roam and don’t want to buy a cage, you can just block off a corner of the room for now and then expand his space slowly.
Oh and don’t force yourself on him. Let him come to you. One of the things I like to do is ignore them. Lay or sit on the floor (to be at the bun’s level) and have a book or something with you and just do your own thing. Eventually bunny will get curious and come to visit. When he does, don’t grab him. The frst few times, you may not even want to pet him. Move slowly and quietly – don’t want to startle him – and let him build up his trust with you. And keep a few raisins or some bits of carrot with you to feed him when he comes over to you so he learns that yummy things appear when he approaches you!
Also… welcome to the forum! Bless you for adopting a bunny. PIx are much appreciated around here.