Wiring – Is this a faulty GFCI receptacle

It certainly seems like you have a ground fault somewhere. Is it possible there is water damage anywhere inside the garage?

Since you've already replaced the GFCI, I think it's safe to assume it's working properly and is tripping as it's designed to. That means somewhere current from the hot or neutral wires is leaking to the ground wire or somewhere else. (I'm guessing it's a neutral -> ground short, since it only happens when you try to use an appliance. A hot -> ground short should trip the GFCI as soon as the power comes on.)

Here are the things I would try:

• Does the tripping happen at the GFCI, or just the downstream outlets? If it's easy to tell which order the receptacles are wired in, start by plugging something in to the first one downstream of the GFCI. Does that trip it? If so, the short is probably between that outlet and the GFCI. If it doesn't trip, move to the next outlet and see if that trips, etc.
• Take the cover off all the outlets and inspect them. Look for any signs of moisture, or a loose wire nut, or a conductor touching the side of the box.
• If you don't see anything suspicious, start pulling the outlets out and rewiring them. Disconnect them all and start adding them back one by one.
• If you still haven't found the issue, it's possible the wiring inside the wall got damaged somehow. Possibly a combination of mice chewing on the wires and moisture from the storm? If you've reconnected the wiring to the outlets one by one, starting with the closets, you should at least be able to tell which section of wall the defect is likely to be in.

Other notes:

• I'm not sure if the continuity tester will help much, the problem is a short not an open circuit. Especially if the problem is caused by a neutral / ground short, those are always going to test as continuous since they connect at the main breaker box.
• I wouldn't worry too much about the voltage readings you got with the GFCI tripped, open circuits can sometimes have small but non-zero voltage readings.

GFCI devices work by measuring the current flowing on the ungrounded (hot), and grounded (neutral) conductors. They do this by running both the conductors through a current transformer (CT), which produces a current on the secondary winding whenever there's a difference in current between the two primary conductors. So as long as both the ungrounded and grounded conductors are carrying the same current, there will be no current on the secondary of the CT. If there was a ground-fault, the current on the conductors would be different and a current would be induced on the secondary of the CT.

If there's no current flowing on the circuit conductors, then there's no way to induce a current on the secondary of the CT. Therefore, there's no way to trip the GFCI. So if you've removed the fixture, the GFCI should not trip.

If this diagram is accurate...

The only way the GFCI could trip, is if current was being introduced somewhere. Like if one of the conductors was shorted with a conductor from another circuit, or a different part of this circuit.

There has to be more to this situation, than the information you've provided. Simple explanations might include:

• If the switch is in a box with other switches, and the grounded (neutral) from the GFCI was interconnected with the other grounded (neutral) conductors in the box.
• A couple cables are stapled together, and the staple has shorted some of the conductors together.

Since you say that removing the white (grounded (neutral)) conductor from the LOAD terminal fixes the problem, I'd guess that the problem lies with that conductor.

If you have a really accurate ammeter, you could clamp it on the white wire and see if there's any current on it. To do this, you'll have to connect the wire to the line terminal (as you described in a comment). You'll only do this temperately, while you're taking the reading.

Alternatively, you could trace the wire and look for damage, or interconnection with other wires.