Hello, Welcome to Just Answer and thanks for your question. BV is an overgrowth of normal vaginal bacteria and it does not cause PID. That is caused by an STD (usually gonorrhea or chlamydia). Yeast infection also will not "cover up" either BV or PID. It is quite possible to have cysts on both ovaries at the same time. You should feel better once you begin the birth control.
You don't necessarily have to ovulate for a cyst to form. It can form from an ovum that is not yet mature, although follicular cysts are the most common type. They form when an ovum fills up with fluid instead of going through the normal menstruation process. Some women have polycystic ovaries with many cysts in both ovaries all the time. There is nothing you can do to prevent them unless you have polycystic ovary syndrome (PCOS). This is a syndrome (a group of symptoms) in which a woman has some of the following: irregular menstrual periods, lack of ovulation (which makes her infertile and increases the odds of more cysts forming), more hair than normal on the face. chest, etc., overweight or a tendency to gain weight, insulin resistance, acne, skin tags, and pelvic pain. Treatment is weight loss is overweight, regular exercise (30-60 minutes daily), and metformin to treat the insulin resistance. You can read more about the condition below.