Stairs can vary depending on whether your bunny likes to jump or change levels, or stays low, like my bunny Samantha does.
But new carpet I’ve had to cope with. I’d suggest going around all of the edges and push the carpet down firmly where it meets the walls. My apartment had the carpet laid down rather hastily, and I actually had to rehammer some of the carpet tacks and get it to stay down so little teeth couldn’t pull it up.
You can also cut thinner strips of sea grass mats and line the edges, like a flat baseboard, so they chew the mat instead of the carpet. A little spray of vinegar and water will not smell good but don’t overdo.
The best way to get a bunny to cope in a new home is to keep their own habitat, and their routine, as much the same as possible. I’ve set up my bunny during a move in his house with all his familiar stuff and meals right on time with a small area of play time with me in the new place at the usual play time. I didn’t have stairs, however.