Welcome to BinkyBunny! 

1. Youve had him a day. You should let him stay in his cage/pen to get used to it before trying to hold him too much. Holding him and moving him around can really stress a bunny out. Dont worry about his sleeping habits until hes much more comfortable and settled in. That being said, some rabbits are very nocturnal. You might see him sleep a lot during the day, but at night he might be crazy energetic. Giving him time to settle in will help with this as he will slowly adjust to your schedule (usually)

2. Depends on the temperature. What temp do you keep it around him? Most bunnies are ok up to mid 70s, ive been told. We keep our apartment 72 or less and have never had any overheating problems. Some rabbits sleep with their eyes open when they are uncomfortable or nervous, but still very tired. I wouldnt worry about this at all. Now with his breathing, you should watch that for a few days. Is there any discharge coming from his nose? If there is, he needs to be seen by a rabbit savvy vet to be tested for snuffles/allergies/ect

3. Do not give greens to a bun under 5 months old. Ive been told it can really mess up their digestive tracts. At his age he should be given alfalfa hay and alfalfa pellets as the majority of his diet. You can slowly start mixing in timothy hay and timothy pellets as teenage+adult rabbits need timothy instead of alfalfa. After hes about 5 months old (someone correct me if the age is wrong) you can start introducing greens. I believe an adult rabbits diet should be something like 75% timothy hay, 15% greens, 8% timothy pellets, and 2% treats

4. Bunnies cant be very destructive. Specially when they are young. Lure him away with food and try to encourage him to only play with his toys. If he has a pen, it might be a good idea to line the bottom with cardboard or coroplast to help protect your floor until he grows out of his destructive habits. Diggy boxes might also be helpful (a box with a hole cut into two different sides; fill the box with shredded paper)

5. We use shredded cardboard from the store as litter. It holds in a lot of the smells and the rabbits dont try to eat it like they do regular litter. We put a layer of hay over the litter. Get a big litter box, rabbits like space when they go. Theyll eat the hay from the litter box but we also have a tin hay holder thing up on the cage full of hay so they eat that more so. Rabbits like to go near where they eat, so keep everything kind of close, but not too close. As for training, a lot of rabbits go in one space/the litter box thanks to instincts. 2 out of 3 of my buns litter trained themselves. For our third, to train him, we would give him treats when he went in his litter box and if we saw him get ready to go outside of his litter box, we would pick him up and put him in the litter box. Warning, hormones can erase all litter habits so you might have to start again after he is neutered 

1. Rabbits do often sleep a lot, less with babies. But it’s largely periods of sleep, play, sleep, play etc.

2. Rabbits do breath quite fast naturally, but if you notice mouth panting or gasping then it is reason for concern. When a rabbit gets sick, often one of the first things you’ll notice is a change/drop in appetite or a reduced number or change in size of poops. Discharge from the nose and eyes is also cause for concern. If you’re not seeing any of those things, I think you’re fine. You can feel the ears if you’re worried about overheating, very hot ears are a sign that bunny could be too warm. Warm ears are typical, but if the ears feel very hot then it’s possible bunny is too warm. If he’s just cuddling with you and has the option to move away but doesn’t, I doubt he’s overheating.

3. There are people who feed their rabbits a hay and leafy green diet, which is great. However you need to be sure when doing this that you are feeding a very good variety and lots of them to supplement the nutrients that he won’t be getting from pellets. Pellets don’t need to be a large part of the diet, only 5%. Bandit weighed 7 pounds and got 2 tablespoons of pellets daily, unlimited hay and an assortment of fresh leafy greens. I say leafy greens instead of vegetables because vegetables is more generic and most veggies should be fed very limited amounts as a treat (carrots, tomato, etc). What I liked about it is that he wasn’t getting many pellets, but I could be sure that he wasn’t missing out on any nutrients. Also just as a side note, when introducing greens, always make sure it’s one at a time you introduce and start with very small amounts to avoid stomach upsets. If he’s not interested, it can take repeated introductions to get them to actually try something.To add on to the above poster, greens can be introduced younger than 5 months, it’s usually more successful if the kits had greens with their mother while they were still weaning, but it can be done. If you are noticing that it is causing stomach upset, wait a couple of weeks before trying again.

4. I can’t really make too many suggestions here, the best deterrent here is really to cover up the spots in the carpet that he’s chewing, with things he can chew such as hay or seagrass mats. If you have towels in his cage, I would remove them. Fleece blankets are relatively safe when chewed on, but the long fibers in towels and other blankets can present a hazard if ingested.

5. You can litter box train for both pee and poop, keeping in mind that a trained rabbit will do about 90-95% of their poops in the box, which is pretty good when you consider how many they do in a day. To start with, set the box up so that bunny has plenty of room in it, and have hay accessible either from a hay rack or in one end of the box. Rabbits like to poop and eat. To start with, pick up all poops and put them in the box, and wipe up any pee messes with a paper towel and put that in the box as well to get it smelling like a place to toilet. If he picks a different corner of his cage as a toilet zone, move the litter box to that corner. Be consistent and with time he’ll get the hang of it, you should be able to get a fair start on it before his neuter, though some are better than others. When you do clean out the litter box, keep some of the soiled litter to put back in afterwards as a reminder that it’s his place to go.

I hope that helps