Hi LeftWing...if your name was RightWing there is no way i'd be helping you Well, maybe I would haha
Anyways, to answer your first question. I'm glad you were able to figure out your exercise routine and assuage your irritation. You asked if it was a reaction from the sweat itself...well, it really varies from person to person (as do most things that have to do with acne!). Your irritation may have been from the sweat, but it also could have been from working out (hormone levels changing, stressing your body out, etc). I would first take care of the most obvious issues (which are overlooked pretty often by people); wear light clothes, stay in the shade when you can, drink a lot of cold water, don't walk too fast to class, and things like that. If you do find yourself sweating, the only option I can think of is to keep a *CLEAN* washcloth with you and dab (gently) your face if you feel sweat coming on. Honestly though, i would first try nothing and see what happens. Our bodies are meant to sweat for a reason, and it may or may not affect your acne/redness.
Question two: you are very smart to be thinking like this. In my experience, BP usually bleaches things the most right after you put it on (and an hour or two later). However, if rain causes it to run down your face there is definitely a possibility that it will bleach your clothes! Solution...umbrella!