Hello Jackson,
Your question is very interesting. There are many different GFI devices on the market and most of them nowadays use auxiliary power to do the current measurement and tripping of the switch/relay that is built into the device. There are also "direct action" devices that need no auxiliary power.
The auxiliary power is always derived from the line and there are three main techniques; transformer, series capacitor and switched power supply. The transformer type has a coil (as you assume) between line and neutral and the 360 ohms mentioned seems to be a reasonable DC resistance if the primary voltage is around 110 V. We usually see around 1 kohm here in Europe, but that is natural with 230 V.
So I guess that the device you measured has a little transformer for auxiliary power.
Regarding the halogen bulb, there is a problem. The filament is usually very close to a short circuit - not open as stated elsewhere. It is also almost always a low volt device (not talking about 500 W lamps, are we?) and thus needs a transformer or some other means of voltage conversion. The conversion sometimes takes place in the socket of the lamp. I think that you have measured such a lamp.
Again, there are different technologies to chose from. The transformer would give you a resistance if you measure the primary side. A swithed power supply with 30 - 40 kHz output has an input rectifier which feeds a capacitor and the switching starts when the voltage across the capacitor reaches about 100 V. Below that voltage, there is no switching - and no power consumption. That is probably why you read infinity when you test the lamp with a low voltage ohm-meter.