The red rash spots aren't likely to be an STD - no STD would present like that. It could be an allergic reaction to something, or a fungal infection, or something along those lines. If that persists, see your doctor.
1. Do I have any risks for any STDs because of the rash-like spots?
Not because of the spots, but because you had unprotected oral. Receiving unprotected oral puts you at risk for gonorrhea, syphilis, NGU and genital herpes type 1 (if you don't already have herpes type 1 orally).
You can test for gonorrhea and NGU (an infection in the urethra caused by anything other than gonorrhea) at 5 days.
You can test for syphilis at 6 weeks.
You can test now with a type specific IgG herpes blood test for hsv1. If it's negative, you can test again at 4 months. If it's positive, it's likely a pre-existing infection which you probably got as a child, like many others. About half the adult population has oral hsv1, even if they've never had a cold sore. 90% will never get any symptoms.
Assuming your partner has a vagina, the chances of you getting anything orally from performing oral sex on a vagina are very low and don't really require testing.
2. Should I go get a full test tomorrow or soon?
You can go tomorrow if you like, if clinics are open near you. Oral sex is lower risk than vaginal or anal sex, so unless you have symptoms, or you are worried about infecting a regular partner, you can wait for this.
3. How many times should I repeat the test to be sure I don't have STDs?
See the above guidelines.
I don't see any reason to rush out and test since your only risk was unprotected oral. If you are concerned about infecting a regular partner, you can test for gonorrhea and NGU now - those are the most likely things you'd get.