Hi and welcome!!!
I will start off by saying Congrats on your little one!! Good news! It seems like he is just acting like a teenager will (eye roll). Most bunnies really hate (with a passion) to be picked up. This is not something we can do much about bc it's a species driven instinct. The best we can do is live with it, and come down to their level for interaction. I only hold my bunny Acacia if I have to groom her or transport her. Everything else is done on floor level
What kind of cage set up do you have? If it opens on the top then that is your number 1 problem. Bunnies have an instinctual fear of being came down on from above.. Why? Because the sky = predatory birds. Even though your bunny has lived inside with people his whole life he has irrational fears. (Kind of like me avoiding entire rooms if I see one spider crawling around in it). My advice on the cage, either get him a new one or (my favorite idea) BUILD him an NIC pen!! You can leave the top open so you can step in and out, you can make it as big as you want to fit your space, you can add shelves and cardboard boxes, the ideas are limitless with NIC cubes. I bought all mine at Wal Mart and in total probably spent about 80$ (that's enough to make 2 huge pens, for when I had Akina) and cable ties (or zip ties) available at the dollar store.
Acacia was the same way, I could not pick her up for the life of me and found I was doing a lot of things wrong. She used to live in a dog kennel with doors on the top and sides, and I realised early on that coming down from above just freaked her out. She stomped her foot a lot. Then when I built my first NIC pens (open top) I went in and sat down in her space. Talked to her, respected her time hiding from me under the shelf in there, never bothered to try and remove her and always just talked to her. Sometimes I offered her treats and she eventually learned to trust me. These things can take time depending on the past of your bun bun. Anyway, Acacia is now 4 1/2 yrs old and best kind. She's not a people bunny but she tolerates being held.
Acacia trusts me, as I mentioned, but is wary on other people too. The reason might have something to do with their past. I know Acacia came from some fool BYB who kept Acacia, her 3 sibings and mama bunny in a cage commercially built for 1 rabbit. She was taunted by kids and a dog before I got her. So same with your bunny, other people who have not spent ample amounts of time talking to him might be dangerous to him. He just doesn't know who they are and what they may do. It's okay, it's normal bunny behavior. Some people want, or expect, a social bunny who jumps into their laps like a cat and who greets everybody with a friendly bunny gaze. This is almost never the case. I know some are apt to prefer human attention to toys or food but it's more common for them not to. Have whoever you live with sit and talk to the bunny to earn a bond.
He eats carpet?? Bummer. Bunnies will chew almost anything. You can block his access to carpet with a vinyl mat (I used to have the same type that receptionists have, it has little nubs on the underside to help it stick to the carpet and not slide around) Those cheap foam puzzle pieces used for nurseries work for some people but I found mine got chewed!! In my old house I lined the carpet where the pens were kept with newspapers and old blankets (in winter). Newspapers are free for me, once the credit is marked off. I had absolutely NO trouble getting them free from 3 different types of businesses. Stores which have a weekly flyer also discard them at the end of the sale, I found most of my stock came from outdated flyers. Bunnies can chew them and be fine.
Your bunny is at the age now where he is experiencing teenager hormones. He might begin to spray soon and he will likely hump things. He can be neutered very easily and this will stop many hormone-induced behaviors. He will probably also become more social and willing to make friends without bunny testosterone
You can find a vet on www.rabbit.org
for your area as well as learn more about the benefits of neutering.
Some simple toys you can try to keep his mind off things like chewing things that aren't his, get an empty toilet paper tube and fold in oth ends to make a closed up ball, fit some treats or pelets in there and make him chew through it to get out the food. You can get a SlimCat ball which does the same thing but is made of plastic. Bunnies love to throw things that make noise (especially at night
) so get him some rattles and interact with him.
Moniter bunny and each time you see him go for something he shouldn't chew just snap your fingers or toot or hiss and tell him no. Don't yell just be harsh. Once he stops, praise him. I found consistancy in the "bad behavior noise" works the best. Everytime Acacia gets into something I snap my fingers and say no. It's worked for years now. If bunny is food oriented, you can train him to do silly things and that helps to distract him from boredom chewing.
Anyway, that's my rambling for now!! LOL It's all over the place but I hope it helps