First of all, congrats on your new bunny.
All of those behaviors sound very normal for a recently neutered bunny in a new home.
If he’s scratching the rug in his cage, I would remove it. You could get some seagrass mats (made for bunnies, check out the Binky Bunny Store) instead.
Bunnies love tearing up cardboard boxes. It’s a fun hobby for them. I let my buns tear it up and I just clean up the shredded pieces every other day. You have to provide some form of entertainment and, for many bunnies, cardboard boxes are a great option.
The litter problems are probably a result of his crazy hormones (they won’t settle down until a few weeks after the neuter) and his acting territorial in a new home. Pick up the poops and put them in the litter tray. If he seems to prefer a certain spot for peeing and pooping, try putting the litter box there. Bunnies will often pick their bathroom spot. Make sure his litterbox is large enough (he should be able to lay down) and place hay on top of the litter to encourage him to spend time in it. Just be patient and persistent.
From my experience, most buns aren’t crazy for wood blocks. Mine do enjoy willow, although that can also be messy.