I may have missed it but I didn't see any mention of gloves.
I don't think they are necessary if you are treating people you don't mind swapping body fluids with. For people other than that (even friends) it can be a good idea to carry gloves to prevent blood from getting into any cuts/scratches or etc on the first aid providers hands.
Nothing to be embarrassed or self-conscious about. It isn't rude. If you feel uncomfortable, tell them you are keeping the wound clean.
Bloody odds and ends can be put in some other plastic of some sort that is probably in someone's pack or burned if not plastic. Prb best to just put it in a bag. It may be a bit much to carry a labeled bio-hazard bag... That's bound to make the injured person feel unwelcome.
Nitrile gloves are pretty sturdy things. Latex, of course. Some people may have thoughts about the type of glove. Subway sandwich shop gloves might be better than nothing but not really the proper thing, I think.