Hello, it sounds like a very stressful situation! Is your bunny neutered?
Some bunnies like to pee on beds because they smell like you and are soft, I would block him from access to your bed. If he’s chewing on cords you can buy some cord protectors. I really recommend cord protectors so your bunny won’t chew a live wire and get shocked! Also for him opening your closet can you try a baby lock? My dog used to be able to open my bedroom door so we got a baby lock so he couldn’t.
For carpet you can put down some fleece blankets and weigh down the corners with tiles. For the walls I’ve seen people use NIC grids and set them up along the wall. NIC grid can also work for blocking off underneath the bed better.
In short the best way to keep a rabbit from chewing something is to make sure it’s out of his reach. Chewing and digging are part of a bunny’s nature.
Putting him back in his cage isn’t going to help, it’ll just make him frustrated.
Getting some toys he can destroy will help divert his attention too! You can try dig boxes, willow balls, etc.. If he bites you I would recommend squealing so he knows you’re hurting him. My one bunny used nip at my hands often so I screamed as if I was in pain(my friends said I sounded like a goat haha) and after about two months of this he slowly learned and stopped. Good luck Amber!!

Unfortunately these are typical rabbit behaviours. Most rabbits aren’t ‘nice cuddly bunnies’ so the first thing you need to do is change your expectations of him.

That said, there is lots you can do to improve the situation!

Firstly, rabbits calm down a lot after being neutered.

Chewing
Lots of rabbits want to chew. It’s your responsibility to move things that you don’t want chewed, or that aren’t safe to chew, out of his reach. He doesn’t understand that chewing is bad, so you simply have to make sure he can’t chew. You can block things off with C&C grids, with cardboard, wire protectors, tiles on patches of carpet that he likes to chew etc. You can also give him safe things to chew in his hutch, like willow or apple twigs, cardboard boxes, and wood tunnels or balls.

Bedding
Lots of rabbits have an obsession with peeing on beds and sofas. There’s theories that they like it because its soft, or because it smells strongly of human. Honestly, the best thing to do is block of access, but if that’s not possible you can try covering the bed with a plastic shower curtain and a blanket that smells like him on top of it. He’s also probably started this recently if he’s around 3-4 months old, as he becomes a teenager.

Biting
Rabbits don’t generally bite for no reason. They’re biting to communicate something. If he bites too hard, give a high pitched squeal so he know’s he’s hurting you, but also think about what you’re doing that has upset him. Eg, lots of rabbits don’t like being picked up.

Punishment

Rabbits don’t really understand the concept of punishment. He won’t understand why you’re putting him back in his cage, he’ll just be upset that you are. You need to redirect his behaviour. So if he’s chewing a cupboard, offer him a willow stick to chew, and block the cupboard off. If he pees on the bed, move him to the litter tray. Spraying him with water or locking him away won’t teach him that what he’s doing is wrong. 

I’m laughing as I read this as my new rabbit was the EXACT same. I have her almost 3 weeks now but if she manages to get on the bed (without her fleece blanket on it) then she can’t resist going to the toilet  It’s so frustrating but the blanket defo works as they don’t feel the urger to leave their mark if they can already smell their scent!! I found lying on the ground so I’m at her level makes her run over to have a sniff and she’s started jumping on me but she’s not exactly tolerating me rubbing her just yet unless I have some treats in my hand…Takes a fair amount of time to gain their trust as they’re natural prey animals. I find she hides under the chairs for hours at a time but as they’re burrowing animals they feel safest in small or dark corners or under beds. I got her a little bunny tent online which she hops into now when she’s not feeling too brave and I made a little den out of a cardboard box with holes on the sides for her to run through and she’s loves to explore and eat it up. Trick is if you have to block something off, replace it with something they can sink their claws or teeth into 

When we got Gary, he tore up two of my dad’s suits, a pair of leather shoes and my school uniform.
We bunny proofed a spare room (took out all wires, sculptures, etc) and kept him in there for a little bit.
Now he likes to stay in there, and won’t chew anything that’s not inside that room, or pee elsewhere.
Of course, he still poops in random places, but it’s much better now.
Hope this helped!

I agree with the others about bunny-proofing. Also, if he isn’t neutered, then neutering would likely help his behavior.

As far as “timeouts” go… I agree and disagree at the same time. So a rabbit may not understand WHY they are being “punished”. That I agree with. But timeouts can be useful sometimes. It has to be done right though and not overdone. Leopold, for example, is deaf. He does not know what “no” means. When teaching him what is and is not acceptable, I had to do a lot of redirecting. However, there were days when he just WOULD NOT STOP. o_O Putting him in timeout for a little while did seem to help. I don’t think he understood that he was being “punished”. But it did interrupt the cycle. So when I would let him back out, he would focus on something else. Like I said though, it something to be done sparingly. More like a last resort kind of thing. I would definitely try redirecting first. Offer acceptable alternatives and direct him to those things when the object of his focus is not something that can be blocked off or covered.