Depo-Provera only. Birth control pills / NuvaRing have return to fertility within weeks of stopping. Many women start birth control pills as teens and stop in their late-30's when ready to conceive - fertility is the same as non-pill users, their age is a factor not the time on the pill. Depo-provera shots can stop periods for 6-18 months after stopping - during this time fertility is low but no long-term bad effect.
Not likely. The hormones in prescription birth control typically clear the body and wear off quickly. Hence, women are advised to use back-up if they miss a pill. Unfortunately, aging and other medical conditions as we age can interfere with our ability to conceive.