The GFCI "stopped working" during the winter. You most likely mean that it no longer provided power. You shorted its wires and failed to trip a breaker. I'm going to bet that the original failure wasn't the GFCI itself, but in the wiring leading to it. If a non-contact tester shows voltage on both sides, I'm going to further make the guess that the neutral conductor is broken somewhere. Your two-wire tester will likely show voltage between hot and a known valid ground. ("known valid", in this case, means "not necessarily the ground wire in the gfci box.")

While it might be argued that #10AWG is the optimal choice to deliver 15 amps over 80 feet, I seriously doubt that this is the cause of your GFCI tripping problem. I ran your numbers through this calculator and came up with a voltage drop of only 4% with #12AWG. (3% is the preferred target, but big whoop.) And that assumes that the pump motor draws the full 15 amps. If it's a 1/2 hp motor, it's probably drawing less than eight amps and even a 1-horse motor will pull a little under 15 amps except during the start-up surge. And even with #10, the surge will cause a significant voltage drop for an instant. I've heard that some types of motors--especially capacitor motors--don't get along very well with GFCI circuits, but this is a pump designed for a pool, right? It almost has to be capable of behaving with a GFCI breaker. And I can't imagine that the weed whacker was too much for the #12 to handle. I'm inclined to go with Henricus's first instinct--that there's a ground fault somewhere. That is, after all, what GFCIs are all about. It could be something like a nail or cable staple that nicked the insulation on the NM wire, or a single errant strand from your THWN that's hanging out. I dunno. This isn't a very high-tech approach I'm proposing, but if you are left grasping at straws, start at one end of the circuit, say the box, undo every paired connection, attach them to something that'll generate a load and find out at what point the breaker doesn't trip under load. Sorry I don't have a better approach. Maybe someone else will.

I assume you mean AFCI, not GFCI, since this a a bedroom. It may simply be a defective AFCI breaker. This has been very common since AFCIs were introduced. First, try a new breaker. If that is no help check with the building dept in your area if AFCIs are even being enforced. You may not even need it. If you are in NYC as your screen name suggests you will likely need it. I believe NYC has adopted the NEC with amendments. I don't know if the AFCI requirement was adopted or not.

The garage GFI requirement is a problem for both refrigerators and freezers -- in fact anything that you need to have powered all the time, but don't check every day. I plug a night light into another receptacle on the same breaker as my garage freezer, one where it catches my eye when I go in the service door. If it goes out, I know to check the freezer. You can also buy power failure alarms -- just search online.