If the pain is just with her period, I would say it's an endometrioma (chocolate cyst) not a regular cyst as they are not specifically cyclic as far as I am aware. I have had a large endometrioma on my left ovary. They are painful during your period because you bleed into it and it gets larger, also the blood that's unable to escape contributes to the pain. Regular cysts normally are painful until they burst, so not just during your period.
If it's normal cysts then I would guess that it's a reoccuring thing, not the same cyst over such a long period of time. And yes, BC would stop the cysts appearing if they are follicle cysts, but like I said I would put money on it being endo related.
As far as fertility goes, the doctor sucessfully drained my cyst without harming the ovary. I know I ovulate out of it as I can feel the pain on that side sometimes.

ETA why they haven't been picked up, well endo is hard to diagnose via ultrasound, endometriomas can, but again, they are hard to see and ovaries sometimes hide too!