WELCOME and congratulations on your soon to be new bunny! Sounds like he will have a wonderful loving home with you already since you care very about how he will settle in.
Question #1: I know that I should just let the rabbit chill the first couple of days, but does that also mean not opening up his cage for exercise? You can open up the cage, just let him come out on his own. Supervise so that he doesn’t get into the habit of peeing outside of his litterbox.
Comments #2 He’ll be staying in a NIC cage, two wide, three long, and either two or three tall, depending on how high his family says he jumps, and for exercise I was planning on letting him run around my rabbit-proofed bedroom. Are you putting a top on the NIC cage or making a pen out the NIC cubes – iF you are making a pen, then clip a sheet to the top to prevent him from jumping out.
Question #3 If a rabbit is litter-box trained and then moves homes, is lapsing a possibility? Yes, it is possible. See where his litterbox is in his current set-up and just try to stick to the same side. Also, add a healthy treat to the box to encourage him to find it and use it.
Question #4 If so, what are recommended floor coverings for NIC cages? I don’t use NIC cages anymore. I used to put sheet vinyl down, but I think others have used Coroplast which works very well. It’s slippery, so you would want to put a towel, small blanket, or something washable over parts of it. Others here who have Cube cages may have more suggestions
Question #5 Can you tell just by looking whether a rabbit has been neutered? … Hopper likes to mount their cat, leading them to believe he is unaltered. Testicles usually drop by four months, so you should be able to see them, however, males can actually pull them back up which makes it difficult to see. So if you are unsure, have a vet determine this. The mounting behavior is not gender related. Though mounting can be for mating, it is also for dominance. So Kale could be just letting the the cat know that he’s the top dog…er…bun in the house.
Last Question: I’m not sure what the family has been feeding him (if they’ve been giving him greens or not), and if they haven’t, what is the best way to introduce the new food to the bunn’s diet? Hopefully you can get details about his diet from them. IF they haven’t been feeding greens, then offer one type at a time. Add a new one each week. Have you seen the list of veggies yet from the diet section? Also, it’s good to know what kind of pellets they are feeding, and if you need healthier ones, then be sure to change that too gradually over a months time. If you need help determining if the pellet is healthy, please let us know. Is he also getting unlimited hay?