For Moen, there is a part called a "posi temp" and this can be replaced to manage the hot water. You may need a specialized tool called a puller. There is an amusing You Tube with Steven Lavmoniere that shows the process to do this.
Maybe. If it is a Moen or other type of single handle shower valve, try changing the cartridge first. If this doesn't work, you need a new shower valve.
ANS2 - Shower valves of all types will clog up with any bits of sediment or rust in the system. This is very common. Take the valve apart and check carefully inside.
Replace the shower valve.
The valve is not be diverting all the water. This valve may need to be replaced.
Yes, remove your old shower valve and fit a new Moen anti-scald valve. This will not change when other items are turned on.
If you mean shower valve, screws are for shower valve coverplate.
you have trash in your line take the shower head off and remove trash or possibly in water valve at shower
Behind shower valve handle and trim plate
May be build up in the valve itself or the shower head.
Though not a plumber, I can say that shower valves are usually made of brass(a very good material). Therefore they are also expensive to replace (50 dollars for the valve, 150 dollars per shower valve if done by a competent plumber). Therefore, when you notice a leak coming from your shower, try to replace the seat washer (usually the hot water side needs to be replaced first). If you still have a leak, it's time to replace the shower valve.
Have you replaced the seats and the washers and stems if required? Have you do it properly? If so you have done everything possible and the valve itself is no good and it should be replaced.
Check the supply valves. Chances are the hot water supply valve is not fully opened.
Cut out the drywall in the wall behind the shower. Replace the valve. Patch the drywall.
A leaking shower valve by installing a new cartridge.