Please don't use a heat mat with a rabbit or anything electrical really. Anything touching the outside of the cage will be inside the cage and thoroughly destroyed by morning. My mini-lop will seek and destroy cords when she roams the room.
I wouldn't recommend dogs and rabbits mixing unsupervised. My buns and small dogs like to sit on the couch together and sniff each other but I'm always right there.
If you want a pair of bunnies neutered males are usually the easiest to bond. Be careful with this un-bonded bunnies can have some really vicious fights. In my rabbit room i have: two neutered males bonded, and two intact sisters bonded, but i also have the third sister by herself because she is picked on by both her sisters and her mother. And then I have two intact male that are separated.
Also, never bathe a rabbit. A sponge bath on a messy bottom is fine. Rabbits get heart attacks and chills when wet all over.
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