Looks like you need a new "branch wye" (that's the name of the piece of pipe with the little arm on it that the hose is connected to). It connects to the "basket strainer" (the piece that goes through the sink opening) using a "slip joint" nut and washer. You should be able to take it off with a pair of adjustable pliers. Make sure you have a new "flange washer" for the connection between the "branch wye" and "basket strainer". If you take this photo to a hardware store or plumbing supplier, they should be able to outfit you with the proper parts.

Yeah, that is just plumbers putty (sometimes people will use it to seal the faucet to sink if they don't have the gasket) you can use a putty knife or razor blade to scrape the rest off of the sink

That's fucking terrifying!

You need to replace that tank ASAP. If the crack opens up while you are out of the house, the fill valve will just continue to run and you will be looking at some serious flood damage.

If ths screw comes up a little bit, use a pair of locking pliers (vicegrips) to grab the head and pull up hard while turning it. This usually works on these pesky floor drain cover screws. Then you can use the next size up thread tap and put in a stainless steel screw (that way it won't corrode).

Note: I saw you mention using a drill to take the head of the screw off. You can do this, but you may need to use a pair of pliers to keep the screw from spinning while you drill it.

The valves are like the most expensive part of the heater. You can purchase them separately and they are fairly simple to install (if you are familiar with plumbing). Considering the age of the heater, you may be better off replacing the complete unit.

Do people not listen to music?

I have no advice whatsoever, just came by to say thank you for the use of the word "blorp" made my morning to say it outloud!

You can pull the rubber guard out from the drain opening and look down inside with a flashlight. It doesn't take much to wedge the blades, especially if the foreign object is just the right size. Typically you will find things stuck under the blade or between the blade edge and the side wall. I have a long pair of needle nose pliers that are curved at the end that I use all the time for this. Good luck.

There are many companies that specialize in fireplace installation. Plumbers can move the gas lines, replace valves etc... but the fireplace company can provide you with the appropriate equipment and perform chimney/vent inspections (which is definitely a good idea with something that hasn't been used in a while).

It may, I usually start out with a bit of that if the valve seats don't look too bad. Kinda hard to tell you without seeing it firsthand. It could be worth a try.

You may have some "pitting" on the "seat" area of the valve body (the whole brass assembly) where the cartridge (which you replaced) comes into contact with the valve body. They make specific tools for smoothing these back out, although often you can get by with some sand paper. If the seat face is not completely smooth then the o-rings cannot seal and you will get water coming past the cartridge.

Is there a threaded plug on the other side of the wall? If so it is probably the end of the line clean out. Looks like it was moved at one point given the cut concrete. You could have a plumber tie onto it below the slab and run it outside and turn it up.

There should be some identifying marks on the underside of the bulb (the sort of gray bit). You might have better luck taking them to a batteries and bulbs store or even a local home improvement store

It looks like the bulb broke free from its threaded base. You should be able to disconnect the power and unscrew the base from the socket. Might need a pair of needle nose pliers to get a grip on it.

They dont make lead based solder here in the states anymore. It's mostly silver, tin and antimony. It's still soft solder on a roll though it doesn't flow as easily as the old lead based solder.

It is necessary because the expansion and contraction of concrete will cause soft solder to separate over time

Just use soft solder and Flux. It takes much lower temps (you can even use a propane torch). Typically silver solder is only necessary for connections that are going to be in a concrete slab or for industrial applications that use high temperatures. And you can pick up a small roll of solder at any hardware store or home improvement center

Go buy a cheap portable bandsaw and a decent blade. You will get way more cuts on things like angle iron. Save the angle grinder and cutoff discs for cutting out shapes, cleaning up corners or cutting off of large pieces.