My gums and teeth really hurt. Pretty sure it's infected and it hurts to eat. Is it because of strep or something entirely different?

Secondary Infection? If you gums and teeth hurt and hurt when eating, you issue is probably related to your gums and teeth. If you have Strep throat, your over worked immune system may allow for other micro-organisms to take hold and cause a secondary infection. Regardless, see a dentist ASAP and get a complete exam and diagnosis. Treatment can proceed from that point!
Strep. Strep throat does not infect your teeth. You may get referred pain. Did the gum and tooth pain coincide with the onset of your strep throat? If so, they may be related. If the pain/discomfort does not resolve after you're over your cold, then see your dentist.
Gum inflammation. sounds like gum irritation/infection. Try gargling with Listerine several times per day, brush and floss after eating. Keep mouth clean, if no improvement in a few days, go to doctor/dentist.