How to I repair this leaking tub valve?
If you have drywall in the shower that has failed due to the moisture you need to replace it.  If the corner is wet and failed you will likely need to replace the adjacent wall too.

At the very least with access in the back you are looking at installing a new valve.  You can troubleshoot yours but for $30 you can get a builder's grade (what you have) or $100 you can get a nice shower/tub valve.  This needs to be swapped out and the connections.  

After you swap valve I would leave that open for a while.  You can inspect the wall and make sure it dried out and you don't have mold growing.  Just know that there is a pretty high chance of this wall failing but really no need to do it now as you should be able to install new valve without hurting it.  

Note:  If you do not want to retile now I would suggest using pex/sharkbite as you can just take them off to install new boards.  I would definitely not weld your copper to valve unless you are doing tile at the same time.