I had a cold prior protected sex & afterwards, cough since. is a month-long dry cough with rare productive clear, white mucus symptomatic of std/hiv?

Not HIV or other STD: No STD causes cough, nasal congestion, or other cold symptoms. Neither does HIV. Acute HIV symptoms, when they occur, are sore throat, headache, muscle aches, inflamed lymph nodes, and rash -- but not cough, runny nose, sinus symptoms, etc. Almost certainly you just have a cold or allergy, not related to any sexual exposure (unless your partner also had a cold). No worries, no need for testing!