>The one with 3 sets has a single wire and a 2-into-1 pigtail.

Okay, that makes sense.

I guess the next thing I'd try to trace out the circuit with a continuity tester. Start with the single wire outlet (end of chain) and figure out where it goes. Just short white/black on that outlet, and test continuity at the other outlets. I'd hope to narrow down where the GFCI load is supposed to connect through this exercise, as well as verifying all the other wiring.

Not that this fixes your problem, but it narrows down the problem circuit. It could be a drywall screw through the neutral, mice, or some other break.

Forgive me if I'm stating something you already know...

I would start at the GFCI. Disconnect the outlet and make sure which wire is live (breaker on). (Breaker off) Connect that live wire to the line side of the GFCI. If it was previously on the load side, it would power the outlet, but not any other outlets connected on the other terminals.

>1 has 3 sets of wires, 2 have 2 sets of wires, 1 has 2 sets of wires)

I'm a bit confused here. Do you mean 3 pairs of 14/2 (or 12/2) coming into one outlet box? Or do you mean 14/3 (black, red, white, and ground)? The former would suggest to me that this box forks out to 2 other circuits. Some pigtailed would be required since you need to send power in 2 directions after power this outlet.

But first, back up a bit. After connecting the 2nd set of wires to the GFCI, flip the breaker on and determine which outlet has power on any of its wires. It may be this one with 3 pairs.