Can I get rid of blackheads completely or is it just something natural that I just have to accept and live with?

Things that help. The best topical remedies for blackheads (comedones) are called retinoids (synthetic vitamin a analogues) the most common ones are Retin-A (and similar products that contain tretinoin), differin and tazorac (tazarotene). They all require a prescription.
Both. 'blackheads' are trapped oil that becomes solid in a pore. They can occur anywhere on the body but are usually most bothersome on the face. Everyone produces oil in varying amounts so no one product or suggestion works for everyone. They are normal and you will likely develop them in the future.