If you are separated she can request from her insurance provider that you be removed from the coverage. If court proceedings are commenced for a divorce the court will likely order her to have you reinstated in that there is a potential to incur a marital debt if you become injured or ill without such coverage and thus incur a debt against the marital estate.
Absolutely. If you are just living separately and not legally separated, you have a couple of options: you can file for legal separation and seek an order for her to continue coverage as a form of spousal support, if the facts entitle you to spousal support, depending on her insurance plan, you can get COBRA coverage for a period of time (but this is not free). If you get divorced, you are no longer qualified for coverage.
Why not? She never had to add you in the first place. Your only hope is to get into divorce court quick and prove that she makes more than you and this is part of her contribution to your support.