No! Per code, and for safety, all ground wires must be connected. If this is n older home with 2 wire wiring then you have no choice. I would suggest the house be rewired for the safety of you family.
Connect the incoming power to the line side of the GFCi outlet. Now run another wire connected to the line side of the GFCI outlet from the GFCI outlet to the switches. Power one of the switches and use that switch to turn on the 2 lights. Run power from that switch to another switch and use that switch to send power to the fan. Mount both switches in a double pole switch box. If the light above the sink has an outlet on it then you will have to connect power going to that light on the load side of the GFCI outlet. If not then just connect it to the line side.
Could be a loose connection somewhere, or a bad switch, or a bad outlet.
Simply run a wire from that outlet to that wall switch. Be sure you use the exact same wire size that you find in that outlet. It will be AWG 12/2 or 14/2. Do not mixes wire sizes. Connect the ground to green ground screw at outlet, and white wire to silver screw, and black wire to gold screw. At the light switch connect all white wires together under a wire nut and push them back into the box. Connect the ground wire to the green ground screw on the switch. Now connect the 2 black wires you have left, power in and power out, to the 2 screws on the switch. Does not matter which wires you connect to the 2 screws.
You can attach/splice into the power feed to outlet (ways of doing this can vary with how the outlet is wired but make sure that all or part of the outlet isn't controlled by an existing switch). You can then take the hot, neutral and ground to a light fixture or to the first switch, then to the second switch. It is normally easier to do this project by taking power to the light (s) first, with 2 wire drops to the switches, from basement or attic.
You need two separate switches. You can get two switches that are in the same form factor as a single switch (2 gang switch). You use one switch for the outlet and the other for the light. You switch the black wires which are "hot".
Sure. For a light you need a fixture to hold the light and the light itself. The light needs to be connected to a voltage supply which you can get from the outlet. In the outlet box you will have a black, white and bare ground wire. Essentially you connect the white and bare wires directly to the light and switch the black wire (hot) through a switch.
The ground wire must remain connected to the box. The frame of the switch, and therefore also the screws, must remain connected to the box as well.
Plug a lamp into a working socket of an outlet to ensure the lamp works. Then turn off the switch and plug lamp into all outlet sockets, top and bottom outlets until the lamp doesn't light. Then turn on the switch and the lamp should light. Often a switch in a bedroom is only connected to one socket in a duplex outlet. It is possible to remove a jumper in an outlet to isolate the sockets for just this purpose. Often an electrician will install the outlet upside down (The third prong pointing up, to identify the outlet.
Connect the incoming power to the line side of the GFCi outlet. Now run another wire connected to the line side of the GFCI outlet from the GFCI outlet to the switches. Power one of the switches and use that switch to turn on the 2 lights. Run power from that switch to another switch and use that switch to send power to the fan. Mount both switches in a double pole switch box. If the light above the sink has an outlet on it then you will have to connect power going to that light on the load side of the GFCI outlet. If not then just connect it to the line side.
No.
How do I connect a wall light switch that has 3 terminals, the third being for the ground wire. Which terminal should be connected to power coming from the breaker box?
Could be a loose connection somewhere, or a bad switch, or a bad outlet.
To control a light with a switch, the switch has to be connected in series with the light fixture.
A single pole switch controls a light or outlet at one location. A 3 way switch controls a light or outlet from 2 locations.
Simply run a wire from that outlet to that wall switch. Be sure you use the exact same wire size that you find in that outlet. It will be AWG 12/2 or 14/2. Do not mixes wire sizes. Connect the ground to green ground screw at outlet, and white wire to silver screw, and black wire to gold screw. At the light switch connect all white wires together under a wire nut and push them back into the box. Connect the ground wire to the green ground screw on the switch. Now connect the 2 black wires you have left, power in and power out, to the 2 screws on the switch. Does not matter which wires you connect to the 2 screws.
You can attach/splice into the power feed to outlet (ways of doing this can vary with how the outlet is wired but make sure that all or part of the outlet isn't controlled by an existing switch). You can then take the hot, neutral and ground to a light fixture or to the first switch, then to the second switch. It is normally easier to do this project by taking power to the light (s) first, with 2 wire drops to the switches, from basement or attic.
Change the wall outlets