Finish the wall first, then put your L brackets on.

A laser level is invaluable for drop ceilings and is the easiest way to keep yourself honest with a consistent height and straight lines.

The ceiling and floor do more to carry the load in a stud wall than the sides do. You can use large toggle bolts into the existing drywall for your top and sides and likely be OK. Use construction adhesive and either concrete anchors or power-driven fasteners to attach the bottom plate to the floor.

If that's a floating laminate or LVP floor in your pic you'll want to take it up in the place where the bottom plate goes so that it attaches directly to concrete and you aren't trapping the flooring down.

Note that if you plan to build the wall laying down on the floor and tilt it up into place, odds are it won't fit because the tilting-up creates a triangle as it rises due to the thickness of the lumber that makes the wall taller than the ceiling as it comes up. Make your wall a hair more than 1.5" shorter and use a double top plate. This also makes the toggle bolt method easier because then you can worry about fastening just the first top plate directly to the ceiling by itself, then slip the built wall in underneath it and nail to the top plate. Use shims to make it tight.

Also double check your height along the entire wall. Never assume a basement floor is flat or level.

The most effective systems are a corrugated vinyl sheeting system that lay parallel across the top of the joists under the deck boards so the "troughs" of the corrugations lie between the joists and the peaks lay across the top. This requires taking up your deck boards and putting them back down, which is really only feasible if they are screwed and not nailed and your deck is not huge. It's a new deck so the fasteners should still be clean and easy to work. This both seals/protects both the tops of your joists and provides a water runoff path.

Next best is a hard lid-type system made of soffit panels attached to the underside. You can use actual soffit material, or corrugated plastic or steel panels that are commonly used for roofing/siding. Metal tends to span longer lengths than plastic materials without sagging. Install with the troughed side on an angle so that any water going through the deck goes into the troughs and then runs off the ends. Depending on how your deck is laid out and how much flow you expect, you may want to put a raingutter at the end to catch this runoff and funnel it aside to a specific point.

I'm looking at this second option as my deck is large and I'm not keen on pulling up the decking boards.

Figure out why they are rotting and correct that issue first. Usual culprits are either lack of proper drainage or mis-aimed irrigation.

That outer area is entirely cosmetic so it's up to you whether you replace it or just rip it out and deal with what's left. It's the casement part around the actual frame of your window that's critical for sealing. The key to replacing those casements will be matching the dimensions of your window, because it's easy to shim a bigger opening but you can't squish the window into a smaller one. The windows are probably screwed to the wooden casements and sealed with clear or white silicone.

The most durable solution to replace what you have would be composite trim. It usually comes in white by default.

My experience with this is that it's almost impossible to manipulate a single wire on a switch without touching the (usually) grounded bracket of the switch, and if that wire is hot, you're getting zapped. In the best case of that scenario it's across your hand but if you're holding that switch with one hand and your other hand touches voltage, current flows across your body. This is how people die. Just turn the breaker off or remove the fuse.

That speaker-and-stethoscope idea is ingenious. But if that doesn't work...

Put some kind of load that is on and drawing electricity on the end of an extension cord and poke it down all the way to the end of your pipe. You need to have something that causes current to flow down the cord. If you have a long enough cord, you could even double it so that you can have the load on the end outside of the pipe.

Call your local utility locator (usually they are free) and have them come with their detectors and mark it. A hot wire in the ground "should" come up on their device.

Leverage beats effort every time.

Use the right tool for the job.

The best multi-function tools are great to have in a pinch and tend to be very portable, but are going to be barely adequate for the tasks they are supposed to cover. Tools designed for the specific purpose at hand will always do the job better.

I'd recommend just cutting the drywall around the tile now and that way you can break it all out in big chunks, and you control where the breakage is. Unfortunately you're seeing drawbacks of a tile job done well.

That looks like the mounting plate for your fan off to the side of your photo. See if the mounting holes will adjust outward past the sides of the box. This box is likely screwed into your ceiling framing with those four silver screw heads that run in a line through the middle of it. Following that line past the sides of your box, use 3" construction screws to run through the mounting plate and between the outside edge of the box and your drywall. They should find the ceiling framing. Mounting like that would be secure enough to hang any ceiling fan from.

You can buy a U-shaped metal sleeve in almost any finish you want that goes over the edge of the door and both knob holes. Look up "door reinforcement plate". I'd still glue and clamp up the edge of that door just to clean things up and strengthen it as much as you can, but for $15 or less this drops right on in minutes.

I was in the same boat as you with our pantry that had a keyed lock. Some kid locked it and the key was MIA, and my door looked kinda like yours after bashing the knob off. The plate covered all the damage and tightened the door up nicely.

It sounds like you're saying your wall is drywall, layer of foam, concrete.

Your drywall is adhered to the wall somehow which means you can use toggle bolts through it to hold your TV. Use a wall mount system that distributes the load by attaching to the wall through a wide rectangular frame (this is a common style), put a big toggle in each corner, and you'll be fine. If your mount is more of a point load, use the toggles to attach 2x4s or a 1'x4' rectangle of 3/4" plywood to the drywall (you could glue it for good measure) and bolt your mount to that. Drywall is a lot stronger that we give it credit for.

We were removing a large commercial duty flat panel monitor (metal case, heavy) held onto drywall with one of these mounts and toggle bolts in a space slated for demolition, and my tech forgot the tool to loosen the mount. Since the walls were coming down anyway, he figured he would just rip the monitor out of the wall. He was a hefty dude but those toggles and mount supported his full body weight plus the weight of the monitor and it took a lot of energetic yanking and shaking to get the drywall to break. When it did finally come down he took a good 2x3 section of wall with him.

The absolute lowest cost material is corrugated cardboard. Large appliance cartons would yield sheets more than adequate to cover the sides of these, and you'd simply tape them to the upright corner brackets. You could probably make "doors" with the stuff too and hinge them with tape. Plastic sheeting is easier to manage, though if it's tearing buy a thicker product and reinforce the "magnet" spots with clear packing tape.

Rudimentary dust management is easier and cheaper than you may think. A box fan with a cheap furnace filter taped to the side of it will catch a healthy percentage of the "random" airborne dust which is what's giving you the most fits. If you are out there while this dust is being generated, you'll benefit from a health perspective as well as cleanliness because fine dust is not great for the lungs.

You can get vinyl/composite trim strips to fur out with. White is the most common color but I'm sure there are other options.