Some people put some hay in the litter box, but the problem with this is that it gets dirty very quickly and so it is wasted which is why most people, as Emmsy said, put the hay either next to the litter box or like me it’s hung slightly above it (from a hay rack) so when sitting in the little box the bunnies can reach it . Be careful of hay racks in case the bunny jumps up onto the top they don’t get hurt or stuck between any wire bars.
And no, indoor bun doesn’t need bedding as such to keep warm but it is nice for them to have the choice and certainly to dig a fleece should they wish. Be aware that some rabbits like peeing and pooping on soft areas, so if you give them a nice fleece or Blankey they may use it as a toilet. Some do some don’t.
Bunnies outdoors would need straw bedding to keep warm in winter.
Most members here use plastic litter trays (like a cat litter tray or big plastic storage box) with paper, or wood pellets or such…. and it’s cleaned out regularly (once a day / 2 days)