As far as the TSA goes, the guidance is extremely clear. You need TSA clearance for training as a private pilot, for an instrument rating, for a multi-engine rating, or for anything over 12500 lbs. You don't need it for anything else, e.g. ASES.
For the visa stuff I can't help. It seems very confusing and I'm just glad that as a green card holder I don't have to worry about it.
n5296s