Aw cute picture. Of course, always be careful with dogs and rabbits. Sometimes, a particular movement in a rabbit do can incite a predator instinct in many dogs, and I have heard of too many heartbreaking stories – so I feel I must warn you to be careful. (i have also seen wonderful gentle loving dogs who wouldn’t hurt a fly and will cuddle with a bunny all day long) So, since this relationship is still so new, just take time to trust it.
The mattress as a floor is a novel idea, and I think it will be very compfy! You will need to watch to see if your bunny ends up being a chewer, and chews through the sheet and the plastic. This could happen especially if the sheet and plastic bunch up in places. So keep an eye on it. Having a couple of sheets to rotate will make it easier to clean without your bunny having to wait.
Regarding cleaning up hay: Regular vacuums really have a hard time with it. I have much better results with shopvacs. You still have to unclog them every once in awhile, but I find it much easier. However, if this sheet method works for you, then you won’t need it anyway – plus the sheet would end up getting caught on a vacuum anyway.
Keep us updated about how this flooring method works for you.