Just my 2cents... I suggest water tight metal flex conduit. Metal flex conduit with a plastic wrap. Probably use four #6 wires, black, red, white, green for a 50 amp breaker.

Troubleshooting is a bitch, only to discover wires were not twisted.

You need to look at the building code before commenting.

I completely disdagree! Use a multimeter to determine which is the neuteral wire. Turn off the main breaker and test white to ground for continuity.

Probably not a good idea then. If the bathroom is on the first story, how are you going to run the fan exhaust duct to the outside? How are you going to run electrical?

Can you cut a hole through the sheathing that is nailed on top of the rafters? Is it the roof deck? If it is the roof deck, do not disturb it. Look in the attic before doing any work.

Another possibility is to install joists and bring the entire ceiling down.

Just a thought... Can you install trim around the fanbox? You can then lower the fan assembly by up to 3/4" depending on the thickness of the trim boards. The only issue would be where the duct attaches to the fanbox.

If you get aluminum duct, you can push it oval so it will fit.

If you do it right, it might look good.

I did not see anything in your post about running electrical and a switch.

I would use reinforced thinset or mastic with a notched trowel . I could see one issue with sticky paper, if you accidentally set the tile slightly crooked, are you going to be able to adjust it? It the tile is attached to a woven mat, I would certainly not use the sticky paper.

Wood to earth contact causes dryrot and attracts termites. The wood on the ground will get wet from the moisture in the ground. Use cinder blocks or bricks.

Easy way... Get a stick, or shim and glue it in the hole with CA glue. Do not fill up the whole with the shim. Screw in the lag bolts back in. The CA glue will strengthen the wood, the stick or shim will make the hole a little smaller so that the lag bolt will be tight. Look up CA glue with activator on Amazon. I use this glue quite often.

Video on CA glue. https://youtu.be/V23s-VQETBs

I heard to use a carrot. Never tried it.

As mentioned before, you need to do more investigation with a multimeter. The red wire (14/3) might go to a junction box and then power the other lights. Test for voltage and test for continuity. Watch YouTube videos on both topics.

Test continuity with a long wire. There are other ways, but for the purpose of brevity, here is one way to do it. https://youtu.be/YLA5u8BJrwg

Get a multimeter before connecting any wires together.

It is really difficult to determine what is going on without using a voltage meter or photos. You really need to do further investigation before connecting any wires.

You need a multimeter before proceeding! You can get an inexpensive meter for less then $20.00.

Check for power between the black and white (should be 110 volts).

Check for power between red and white (no power or 110 volts??) If there is no power, there is a good chance that the wire is charged by the activation of the sensor. If the red is 110 volts then further investigation will need to be done. You will need to find out where the wire originated.

Check for power between the black and red. If it is 220 volts, then there is a separate breaker for the red wire. Just CAP the red wire until you are able to determine what it is for. I highly doubt it will be on a separate circuit (220 volts) for exterior lighting. The wattage is low for lighting. There could also be 110 volts or no voltage between red and black. 110 volts would indicate that a light is already connected to the red wire. No voltage would indicate that both red and black have power and are on the same busbar on the electrical panel OR the red wire is disconnected. If there is no voltage between red and black, check again between red and white. If there is 110 volts, then red is on the same busbar as black. They most likely would be on the same circuit.

Open up the other lights to determine if the red wire connects to that box. Check for power in the other boxes.

You should educate yourself on basic wiring by watching some YouTube videos. Find some videos on troubleshooting wiring.

This is what I suspect: Black wire (line)- hot 110 volts. White wire - common. If there is no power between the red and the white, then there is a good chance that when motion activates the sensor that charges the red wire (load). That powers the other lights.

Provided the red is the load: To install another sensor further on the branch circuit, connect the red to the black together. In this situation, if you connect red and black together, you should have constant power for the other lights. Then you can install motion sensors in the other locations.

Installing a window is probably best.

However, to patch it is going to need some work. I do not see any outside photos. My biggest concern is the opening to the elements. Install stucco or sidding on the outside. Install insulation in the wall. Foam board or fiberglass insulation. Sheetrock, tape, texture, prime, and paint the inside.

Edit: Make sure to take out the wood molding in the pic. When installing the sheetrock, make sure it is flush with the rest of the wall so it will look smooth when finished as if it was never patched.

Great advice! One thing to add, it appears to be a 3/8" hose connection to the valve.

It also might be a fill valve issue in the tank. If it only fills partially up, have you adjusted the float in the tank?

That is correct, but with the cost of wire, 14/3 is avoided.

You cannot add a second switched outlet to a wire that is already on a switch. Then both switches would have to be on for the light to work. The only exceptions would be a three way or four way switch.

You clearly do not understand wiring and should not be messing with electricity.

Your diagram is not correct. You probably should not be doing any wiring if you do not know what your doing. The switch connects to the hot wire, then connects to the light. All of the common wires are connected together.

The power does NOT go to the light then to the switch. That would be a code violation. You would never be able to disconnect the power to the light.