In response to your questions:
1. There is no formal grace period - you lose your status as soon as you lose your job. Although sometimes, you can get away with a small gap like this, you should be ready to go back and get your F-2 visa overseas.
2. The application process takes a few months.
3. Theoretically, yes. This is a very complicated scenario, however and you do need to have an attorney assisting you in the process.