If it continues, I’d get him back in to be seen by the vet. It could be the beginning of an upper respiratory infection.

Otherwise, it could be caused by overly dusty hay or another irritant in the air.

Just like us, rabbits will sneeze on occasion, but if it’s constant and coupled with discharge from his nose or eyes, he should be seen.

My sneeze prone girl sneezes 1-2 times a day, but when she gets a URI it starts to become more and more common and they’ll come in quick bursts of sneezes, the bursts being 3-10 sneezes in a row, multiple times a day. It’s hard for a vet to detect a URI until it causes some sort of irritation, or discharge, so they could have missed the onset at his checkup. You could call and ask the vet what they think, since they just saw him.

My boy Lionel has chronic sneezing and he does this occasionally-but it is usually a URI. If it continues, I would definitely get some ABs from the vet to clear it up.

I wouldn’t be very worried about the digging but I would be worried about the sneezing. You say his nose is damp so this can be the signs of an upper respitory system. If this keeps happening and he keeps sneezing you should probably get it checked out at your vet.

Mine will dig at my pant leg or shoe when they want something. Usually attention or food.

I wouldn’t be terribly concerned about any bugs unless you have spiders with potent venom (i.e. black widows, brown recluses). Obviously you don’t want bugs crawling through your bunnies cage or on their food, simply because it isn’t very sanitary. Perhaps you could call an exterminator and they can recommend some methods that wouldn’t cause a problem for pets.

Bunnies like to dig. That’s what they would do in the wild, it is just natural. My boy Lionel does this too. You just have to redirect them.
I wouldn’t worry about the bugs as long as it’s only the one incident.