Looks like everyone has given you great information. Here's some more info!
EDITED TO MAKE MY THIS READABLE - What a mess. I'm need more sleep.
SLEEP STUFF: As far as how many hours a rabbit gets of sleep per day...that's a good question, so I asked that on etherbun, and will ask my vet next at Rucy's next visit. I'll keep you updated.
Most sleep and rest during the midday and sometime in the middle of the night. Though rabbits are crepuscular (active dusk & dawn), , they can actually become accustomed to your own schedule. I think the reason for that is because the whole reason they are crepuscular is because that is when they are the safest, and what their vision is best suited for. So it's the best time to get up and do bunny things, like eat, play, mate etc etc. But in our homes they can feel safe, and depending on the home, inside is a bit darker then being exposed to the big bright sunlight. So, though they will remain dusk and dawn active, they can adjust that.
Bunnies can sleep with their eyes open, as well as while they are sitting up. "Looking" like they're alert, and being ready to run, even when they're asleep, is important for a prey animal. However, my Jack, sleeps with his eyes closed and usually is fully flopped. Bailey is more of an "alert" sleeper (since she's queen and feels she must gaurd her palace from "evil" Rucy (her nemisis). But even Bailey is beginning to sleep in a more relaxed position.
LITTER TRAINING: It can take over a month for a bunny's hormones to die down. Also, do you have any other animals? If not, then he is still making this place his home. If what you are saying is he poop the piles in the box, and pees in his box, but just leaves random poops everywhere in his pen, then he is still establishing that it is his, and has nothing to do with not being littertrained.
There are things that can help you reinforce that his pen is HIS place. (you may already do these things).
- Let him come and go on his own. If he is not willing to go back on his own then you can put him back right in front of the entrance and not allow him to escape anywhere except for into his cage.
- FOR NOW, try to resist reaching into his cage while he's in there. Clean and refill hay and food when he's out. One he feels he's gotten the message to everyone that this is "his" place, his territorial poops should lessen
- Again, as his hormones die down, this should also get better in time.
If he is doing this around your house, you may need to do what others suggested and confine him more for now - if he has just a cage, you may want to get an xpen to allow him limited running space.
BRINGING IN A COMPANION: How long has it been since the surgery? (everyone has really offered good advice here too)
MOLTING: This actually happens four times a year, two light, two heavy, however the first year of a bunny's life you may see some extreme molting - usually the first is the heaviest. Some bunnies have explosion molts where the hair can just easily be pulled out, and it happens over just a few days, while others gradually do it, and it can take weeks.
Other Oddities: Now we get into our bunnies'' individual personalities. That's the fun part! The fan though makes sense - Bunnies cool off by dilating the veins in their ears, and when wind passes over it, it cools the blood in their ears which then in turn helps cool off their body. So i bet he is enjoying the cool breeze (especially if he's molting - he may have some extra fur that is keeping him warm) Be sure to cover the electric cord. Many bunnies will chew on cords which can obviously hurt them if it's higher in voltage (even if it's alternating current like most homes)
Someone mentioned tooth grinding somewhere, and like everyone said, a soft tooth grind is like a purr. If it's very loud, then it means aggitation and/or pain. But it really is very loud in comparison to the softer tooth grind (they also have sort of a blissful look on their face while purring)
I know I've posted here sort of late in the game and things may have improved already - How are things going now with litter training.