can these STDs cause a swollen cheek and rashes on the face?
Chlamydia could perfectly infect the eyes, and it's actually a really common infection in some places along the world like Africa, Asia and the South East of Spain, which is called trachoma. It's normally transmitted to babies when they are born, because of the contact between their eyes and body fluids from their moms in case they are infected.
Regarding how long can take to develop an infection, that will depend on some many different things. STDs are known sometimes because they can hibernate inside the body for so long without giving any infection.
Gonorrhea is spread by sexual fluids and secretions, the same as chlamydia, so if you are worried of your eyes have been exposed to sexual fluids from another person, you could test for these two three weeks after the exposure.