We noticed water beginning to mist out of what appears to be the supply line (the braided line, not the copper line). That same braided line is connected to the line in the wall...and the top of the water heater. The supply line (from the wall) has a valve with a red knob and I turned it off. Then I shut off the gas, and pilot, opened the hot water in the sinks and the relief valve on the side (rather the TPV). I'm not sure but I think all I need to do is just shut off the cold water at the curb, drain the tank using the sinks, and change the supply line ...is this the right path to take here?

might need a filter sounds like calcium

First thing did you open all the shut off valves in the water lines going to or from the heater. I shut one off one day and forgot to open it up. Frustration was the word.
Check for air locks in the tank and or pipes.
Do you have galvanized pipes or copper? Old galvanized can plug up with rust. Is there a filter in your system?Filters can plug up.

Look at the piping to each fixture. If they are flexible lines you can easily swap them after turning off the water supply. The fixtures themselves are the same so that is not a problem. You can choose to make the swap at the shut-off valves or at the fixture which ever is most convenient. If the lines are the solid copper type you can bend them around to do the job if they are not to short. If line length is a problem I recommend replacing one or both with the flexible type from Home depot or wherever. The type that looks like braided stainless steel is best. Rubber hoses will end up making your water taste funny.

Turn off the water supply from the heater and at the main water supply line outside. place a container under the sink where the flick mixer is mounted to catch the water that will come out of the lines when you undo the fittings.
there should be 2 flexible braided lines coming from the mixer to the water supply lines, undo both fittings from those lines, then undo the brass nut that fastens the mixer to the sink, you may need a 10 or 12mm socket with an extension bar to get up under the sink to be able to undo the nut.
there should be a metal horseshoe shaped retaining plate that holds the mixer to the sink which should come off when the nut is undone.
You may or may not have to remove the flexible lines from the mixer which you should be able to sort out when you get to that point.
I hope this info helps you with your problem, the info may not be exactly how I have explained but that is how most mixers are fitted or removed.

Youtube offers many videos of the water heater installation. Drain first with a garden hose. after turning off water and unplugging electric or shut off gas.

If changing to a tankless, remember they often require a larger vent pipe to the roof. 4" usually, not the 3" for tank water heaters. Local building codes may require a permit and a larger Gas feed pipe.
Also if you're going to use the old Pressure Relief Valve DO NOT test the valve by activating the little lever. Once used over time their internal seals weld onto the closed seat and tend to rip and leak if you test the valves function. Replace with new ( $15) if it's been operated. Most plumbers now use Braded Feed Lines from/to the Water sources, making future replacements easier. You might find the replacement Tank is not the same geometry as the old one, having longer braided feed lines helps when they are on opposite ends of the tank top and the original feed lines are copper and wont reach the new tank pipes or the tank gas starter door is on the back side requiring you to rotate the tank 180. If you're changing size, measure the width and height. It might not fit the old installation.

The thermocouple is the copper one the other is the gas supply to the burner check your flue pipe for restrictions see if a flame from a lighter sucks up the flue from the top hood you may need a new gas control valve

you will have to get a replacement from the manifacture,, they will send it to you, call them tell them wich one it is hot or cold they may have diffrent threads then again they may be the same,,,can you get the threaded part out of the faucet, you will need an extractor, you can get this at a hardware store, it has a rubber seal so you can only tighten it so much, get it ot try again but be gentle,, one thing I found after doing plumbing for 34 years if you snug it and it leaks you can always snug it some more if you over tighten it you can't do anything but replace it,,,, good luck let me know how it goes

This faucet is the hot only version so it has two copper lines and one Santoprene rubber line. The copper line directly under the handle is the inlet from the wall - the second copper line attaches to the hot water tank by the corner. The Santoprene line attaches to the middle of the hot water tank. This faucet cannot be installed without the hot water tank.

If the water heater is working fine than I would not worry about the function of the water heater. What the problem may be is that some of your WATER LINES HAVE MOVED, AS THEY WERE NOT SECURED PROPERLY. The sound you may have heard was the Copper or PVC(Plastic) lines just getting back comfortable. I Suggest checking that your water lines are secured about every 3 or 4 feet with the proper ancjhor....Metal for metal pipe, copper for copper pipe, etc. Sometime as in my house, pipes are run through a hole drilled through a stud or a piece of wood, usually when hot water is run through this pipe. You can here an expansion of the water line interacting with the pipe expanding, if the hole in the wood was not drilled big enough......Experience of remodeling a very old house.......Joe