1) what are the compelling reasons? What do you need the reasons for? extending your I-20? just get your advisor to fill out a standard form that you need more time and take it to the International center. Yes, a minor or a second major seem like a good reason to me
2) no, you can only go to school part time while on OPT. if you start a new program or go full time, you forfeit your OPT. You have to apply for OPT BEFORE you graduate, since it takes about 3 months to get the work permit, and the start of the OPT needs to fall within 60 days grace period after you graduate.