1. You're going to have to advocate for yourself here. While TSA's in theory deal with medical equipment all the time, they can be remarkably callous. Request that the pat down be done in the private area BEFORE they start patting you down (unless you're cool with the TSA exposing your leg bag for all to see which you might be). In theory, they should offer this option but it's pretty hit or miss.

2. Flying with a catheter is no problem whatsoever. I don't personally have a foley (I do intermittent caths) but a few girls on my team do, and they are perfectly fine. In fact, they've kind of got the best deal because they don't have to deal with transferring into the tiny bathrooms/ aisle chairs in flight etc. Many, many paras and quads fly with foleys all the time. Just make sure that there isn't a kink in anywhere when you sit down if you plan on being in one spot for a while.

You'll be fine. Just remember to hydrate well so the risk of infection isn't increased - but that's standard foley info, not specific to flying. Safe travels!