Could a uti cause my genitals to be swollen and itchy, but not cause any pain when i pee?

Yes: This sounds more like a yest infection than a uti. Try topical antifungal cream that you can buy over the counter first. If not better in a few days go see your doctor.

Possibly STD: Symptoms likely due to yeast infection or possibly sexually transmitted chlamydia or herpes. Suggest you have it checked out by your primary health care provider or gyn dr.