If this is anything like the valve I've worked on:

1. Shut off main water supply
2. Block drain to prevent losing parts
3. Remove knob (you've already done that)
4. Slide off sleeve from body of valve
5. Pull retaining clip exposed by removing sleeve
6. Partially reinstall screw from knob into valve core
7. Pull valve core straight out, make note of orientation
8. Clean, inspect, refurbish valve core. May need to replace o-rings or replace core entirely.
9. Lightly lubricate the o-rings with plumber's grease
10. Reassemble

I had a similar problem with my shower valve. Turns out I didn't need to replace the whole thing, just the core. Ordered a new one from Amazon for like $20 and that fixed the problem.

The new core wasn't a perfect fit, so I had to take apart the old one and use some of its parts in the new one (they fit perfectly). It was just a matter of unscrewing stuff and screwing it in, so it wasn't a big deal.

Sharkbites are an option if you don't want to do any soldering to convert to PEX. They're expensive but if you haven't soldered before and it's going to be sealed up again, I would trust them over a globby amateur solder joint.

Key is to clean the copper side up very well so there isn't a burr or old solder that breaks the seal.

I was in a similar boat a number of years ago with the downstairs bathroom, what with no access panel and all. Based on that experience, which was probably 10 years ago or so at this point, I'm pretty much all in on Sharkbite fittings. That, or if you're using copper you can use those crimp fittings that are all the rage these days. I'm good at soldering, but also: fuck soldering. Depending on the orientation it can be a real pain in the ass, and sometimes there just isn't enough space to work in to feasibly do what you have to do with solder.

I was initially skeptical of the longevity of Sharkbite fittings but I've had a pair of 'em in my wall all this time without any incidents to report.

Mine was a mega builder's grade three hole arrangement for which rebuild parts are flat out not available and which was largely apparently made from chrome plated plastic. So I replaced it with a similar but rather nicer assembly that's at least from a company I've actually heard of (I think it's a Delta, but don't quote me).

When you're hacking your access hole in the wall, take measures to avoid also cutting into the pipes in the process. Ask me how I know.

Sweating pipe is fun to learn if it's not an emergency. You'll probably get it wrong a few times though.

The bigger issue here is that the cavity isn't sealed so this is probably the tip of the iceberg in terms of the water damage you're going to find. Modern waterproofing systems have a rubber gasket which seals around the rough opening to divert water, but this goes under the tile.

Higher a plumber? A fair price about 700-1500USD$ and it'll be done in 3 hours. Recommend a Moen Positemp rough in valve as the have easy handle extension kits for just this type of replacement work to account for varying wall thickness. Recommend installing ball valves in your closet and keeping it an access panel. At this point you can install the valves first and then you have everything isolated in case you get into trouble.

That looks like a real degenerate operation. Wouldn't recommend repairing.

If you end up using PEX then it needs to be PEX-A (apparently) for the upper shower bit because the PEX-A fittings preserve the internal diameter of the pipe. The PEX-A tools are more expensive than the PEX-B tools.

Use Apollo brand. Use the Apollo crimping tool: https://www.homedepot.com/p/Apollo-1-2-in-and-3-4-in-PEX-B-Crimp-Tool-69PTKH0014C/301573902#overlay

I bought the tool, used it about ten times and then returned it for a refund. Just be careful with the packaging.