Question #1 is simple: Yes.
She got out of bed.
Yes, that can mean that she got off her bed of laying there for some time.
Question #2 is the tricky one.
Answering questions that ask, "Can you do X in English?" is difficult, because there are several factors at play. Depending on what X is, the answer is often:
1) Yes, you can – but that doesn't mean most native speakers would.
2) Yes, you can – but typically only in certain contexts when it makes sense.
I'm playing around with some sample sentences in my head, trying to figure out when a determiner would sound natural, and when it would sound out-of-place.
Here's what I've come up with:
She tossed and turned for at least an hour after she woke from the disturbing dream. Finally, she got out of bed and got dressed. (Sounds fine)
She tossed and turned for at least an hour after she woke from the disturbing dream. Finally, she got out of her bed and got dressed. (OPINION: The word her isn't necessary, but it doesn't sound terribly awkward, either)
She tossed and turned for at least an hour after she woke from the disturbing dream. Finally, she got out of his bed and got dressed. (OPINION: A determiner might actually be useful if a writer wanted to emphasize that she wasn't sleeping in her own bed)
She tossed and turned for at least an hour after she woke from the disturbing dream. Finally, she got out of the bed and got dressed. (OPINION: The word the seems like its just begging to be removed; if she were in her own bed, I would omit it, although it might be okay if she was, say, sleeping in a hotel room)
Also, determiners are actually necessary when preceding an adjective modifying bed:
She tossed and turned for at least an hour after she woke from the disturbing dream. Finally, she got out of the messy bed and got dressed. (Sounds fine)
She tossed and turned for at least an hour after she woke from the disturbing dream. Finally, she got out of messy bed and got dressed. (Incorrect; a determiner is needed here)
1From your well-arranged account, I get an idea that if there is already implied definiteness, we need not use some article that would be “redundant”. (“The interpretation of bare role NPs invariably definite - CGEL,p.409”) Thank you for your refined answer. – Listenever – 2014-06-17T09:24:10.253
1What was she laying in her bed? Eggs? I suspect that you mean *lying* on/in her bed. – Scott – 2014-06-17T18:14:23.683