That behavior sounds normal. That was exactly how my foster cat acted when I brought her in the house. I think though that it would be better to leave the bed out for the cat though. I know you want her to be comfortable, but I think putting it under the bed will make her think she is supposed to stay under there. Just give it some time. If she really is uncomfortable on the floor, she'll come out.
And I was speaking to someone from the shelter my cat came from, and it could take a long time for the cat to feel completely comfortable. They even said 6 months or more if needed.
Could you ask your friend about the biting? My sister's cat does that when you are playing with him, and he gets too excited. Another cat at the shelter would bite too if he wanted attention. It isn't a hard bite, but more like a love bite.
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