Electrical – Regarding GFCI outlets

A GFCI is only measuring currents and does not care about the voltages on either of the two wires going through it.

As long as the net current is always zero, it really doesn't matter how it is switched on and off.

The real problem with a "switched neutral" is that the contacts inside the fixture are live with respect to ground even when the switch is off. If you should accidentally touch the contacts while grounded, the GFCI will protect you by tripping. But if the fixture were wired correctly, that particular risk wouldn't exist at all.

OK first of all you cannot determine whether the box is properly grounded just using a multimeter. The fact that it reads "0V" does not mean it is properly grounded. The only way to determine that would be to confirm that there is a proper ground wire all the way back to the panel, or to confirm that grounded armored cable is used. In your case I would just forget about the ground and disconnect it, you don't need a ground for a GFCI.

Second, try to get the GFCI outlet working without the downstream outlets hooked up to the LOAD side. If you can get the GFCI working by itself but it trips when you hook up the downstream wires, that probably means there is something downstream creating a ground fault and the GFCI is tripping (as it's supposed to). A common cause might be a neutral wire touching a metallic box or a "bootleg ground" where neutral and ground are connected.

If the GFCI doesn't work even on its own, it's probably defective.