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

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.
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.