Yes birth control should get rid of the cyst. I would suggest taking it so you don't have to experience the pain of it rupturing( its worse then the pain your having now) you should always use a back up method to the first month of pills taken. If you decide to go back on the Bcp please make sure to use other protection as well since it is your first month and the pills weren't taking daily as prescribed.
Birth control can help cysts disappear, yes. And yes, you can still get pregnant. If you're not using protection of some sort, begin immediately. At the very least, use a condom.