I am sorry to read about your mother having difficulty after breaking her hip. As I have seen in my years of practice, when elders break their hips, it is
often very difficult for them to survive and recover.
From just reading your letter, it is difficult for me to say if she is in a coma. But, it definitely sounds as if something has happened to make her unresponsive. I would recommend asking the doctors or nurses in the Rehab facility what is going on with her. Perhaps she has an infection (like pneumonia or UTI), or perhaps she had a stroke. Regardless, they need to communicate to you what happened, and what your options are. If there is nothing they can do to help her, then they need to do their best to make her as comfortable as possible.
Regarding your question about "biting" of the swab, I have seen that behavior before many times. It usually represents a reflex from you stimulating the lips with the swab. Like you, I would not force the swab into her mouth- just gently moistening her lips should bring her comfort.
Your last question is about how long can people survive without food and water. People who are not drinking usually can only survive a few days to a week. If your mother is in the end of her life, please do not worry about her lack of drinking and eating. This is often our body's way of shutting down, as our needs for sustenance diminish. Think about when yourself, when you have been really sick in your life. Oftentimes, food is the last thing on your mind. What really is important is making sure that she is comfortable and not in any distress from pain.