Help with my rabbit, please?

Alright, I just got a rabbit yesterday, her name is Guinivere but I just call her Guin for short. So I'm incredibly confused right now, I keep finding all over the place that my rabbit should have a litter box. Is this true? And all help I can find is based on cages that have wire mesh on the bottom but mine isn't like that! It's like a HUGE hamster cage, (Trust me, she has plenty of space, 6 feet long, bout 5 feet high) with a plastic bottom and I put a couple of inches of hay at the bottom. I'm replacing it tomorrow with a layer of newspaper and then the hay. But, is there something wrong with that? Is it absolutely necessary for her to have a litter box? I know their very clean animals...also she won't come out of her cage, even when I try to coax her. I'm giving her time and space, so I'm trying just every couple of hours. But she won't come out...also I think she's been regurgitating her food. I know rabbit's don't throw up but next to her food bowl is her food in a pile of mush, it appeared after she made a couple of weird noises. I heard this is normal, that they're recycling but I can't find out much about it! Please help, I want to give her the best and loving home possible!