Lexi01, pretty much anything that was once alive can be composted. You do have to pay some attention to the mix of browns (dead leaves, paper, dry grass, etc) to greens (fresh grass, kitchen trimmings, etc.). If you get too many greens and not enough browns it will start to smell. Just search composting and you will find a ton of info. It is not nearly as hard as it sometimes sounds. It does not have to be mixed or "stirred", but it will break bown faster if you do. You also want some moisture in there, but not too much. If you maintain a good balance it will never smell. I have a couple of forums I lurk on just to get ideas on things to compost. There are all kinds of thing sout there that I would never have thought to compost otherwise.
Yes, wood or paper based litter, hay, poops, pee - they can all go in the pile or directly in the garden to compost. I found when I added the entire contents of the litterbox to the garden I ended up with too much nitrogen in the garden. That left me with really nice plants and almost no fruit. So, now I put the poops in the garden and the hay and litter/pee go in the compost pile which helps to even out the nitrogen content in the finished compost. However, there are exceptions to that. Corn needs lots of nitrogen, so for a few weeks prior to planting last spring I added the litter/pee mix to the area I wanted to grow corn and mixed it into the soil. I had the best corn! For tomatoes though, too much nitrogen gives you huge plants and no tomatoes.
As far as winter gardening in Utah, with protection (mini greenhouse) you probably can. I know there are folks in Maine that have fresh salad greens way into winter if they protect the plants.