First,It is absolutely possible for a man’s semen to transmit an infection to his partner’s eye if he comes on her face and some of the semen gets into her eye.
Confirm the symptoms:
Red, itchy, swollen or scratchy eyes
Sensitivity to light
Pus or watery discharge
1) If I got eye chlamydia...is that transferred to my genitals? AKA is it considered an STD that is contagious to a sexual partner? Or is it considered just an eye infection?
Don't worry about transfering to genitals, it is an eye infection if it is.
2) Am I at risk of having contracted chlamydia by having his semen on my face/lips and possibly (but unlikely and if so, in a TINY amount) in my mouth?
Things you need to know:
Wash your hands often. Do not touch or rub your eyes unless your hands are clean.
Never share eye makeup or cosmetics with anyone. When you have conjunctivitis, throw out all eye makeup you have been using.
Do not share towels, washcloths, pillows, or sheets with anyone. If one of your eyes is affected but not the other, use a separate towel for each eye.
Use latex or polyurethane condoms during foreplay and every time you have vaginal, oral, or anal sex. Have just 1 sexual partner who is not sexually active with anyone else.
If you are pregnant and have a chlamydial infection, get treatment for it before your baby is born
Wish you the best.