Let's address #3 first...FIX the toilet! Either the fill valve is not shutting off and water is going out the overflow tube into the tank, or the flapper valve is shot, leaking water into the bowl. The fact that this is getting hot water makes it a double-whammy, wasting water, and energy to heat it. It's getting hot from (likely) that shower valve, but it might be another single-handle valve in the house that has an internal leak and generating the cross-connection. To help isolate where the problem is, wherever you have a single handle valve and have shutoff valves to it, shut the hot off. Then, after purging the lines, see if the cross-connection is now gone. If it is, you know that valve has an internal cross-connection and needs work (often a new cartridge). I'm not very familiar with identifying various brands, and can't tell you how to fix that one or even if parts are available. If you can get to the back of that valve (say from a closet or something), I'd be more inclined to just replace the whole thing with one that will meet current codes. You can legally repair an old one if you can find the parts, though. PBlaster is a spray penetrating lubricant that might help with the screws. Don't try to loosen them without a big enough flat-blade screwdriver, though. It should fit in there tight and be at least the full width of the screw to have the best chance of removing them without destroying them. Most people use whatever screwdriver they have, and then muck up the head - it's sort of amazing how well it works when you use the right-sized tool!