I have a hole/lesion growing in my gum above my tooth, white pus. also, skin eruptions around nose & chin. this started after a bad cold. std?

Tooth abscess ? : The gum lesion is likely part of a draining abscess, for that you need a dentist and/or an oral surgeon, the skin eruptions on face and cheek are probably due to acne- treatment is daily cleansing, benzoyl peroxide, and if severe, retin A (tretinoin) from your dermatologist. If you are worried about STD will need a full exam and serologic(blood) testing with the appropriate screening tests- per your GP/ER md

Oral herpes?: One possibility is new infection with herpes simplex virus, i.e. oral herpes. This is even more likely if your "bad cold" symptoms included severe sore throat, swollen glands in the neck, and maybe fever. Most oral herpes is due to HSV type 1, usually not sexually acquired. But if you performed oral sex on a person with genital HSV2, it was sexually acquired and considered an STD. Get treated!