If you are referring to lamp cord type wire where both wires are brown then yes, connect the wire with the groves to the white neutral and the smooth wire to the black hot wire.
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Because the white wire on a 120 volt circuit is the neutral wire that is connected to the silver screw on outlets and switches. It is connected to the neutral bar in the service panel.
Based on your description it is very likely that either the red wire or black wire is switched and the other is on all the time. In most cases the light of the fan, if it has a light, is connected to the switched wire and the fan is connected to the unswitched wire. This allows you to turn off the light while allowing the fan to remain running. There is no way to know without testing which wire is switched. The white wire is the neutral, connected to the white, or neutral, or common conductor of the fixture. There should also be a bare or green wire in the box to connect to the ground wire of the fixture.
The neutral wire and power wire are never connected together.
A GFCI receptacle can extend its protection to regular receptacles connected to the output side of the GFCI. Each actual GFCI receptacle should be directly connected to a breaker in electric panel.
In the US, house wire has the BLACK wire connected to the HOT phase of the service input. The WHITE wire is connected to the NEUTRAL of the service input. Normally, both BLACK and WHITE wires are each connected to the 2 LOAD terminals. It is also normal that the WHITE service wire is connected to the wider spade outlet receptacle. The BLACK wire would then be connected to the shorter or narrower outlet spade receptacle. In wired screw lamp circuits, it is normal to wire the BLACK service wire to the center pad of the lamp socket and the WHITE wire to the screw shell of the socket. In switch circuits, it is normal to wire the BLACK service wire to the switch. The HOT service circuit is then opened or closed by the switch. Measuring voltage on your neutral means there is a break in neutral. If the neutral is broken you will measure voltage across the break or from the break to ground. Under this scenario you will measure the same voltage as you do on your hot wire and your load should not be working. Measuring voltage on your neutral less than on your hot wire may mean you have a problem where your neutral is supposed to be bonded to ground in your main service panel. In this scenario your load may be working but you need to check the bonding connection.