As said above, with the pump on, open the pressure release valve. You should get a lot of water coming out. Leave it open for a few seconds to ensure the flow continues. If you do not have good flow there, I would assume part of the broken valve may be in the section between the new valve and the tank blocking water flow. Easy to check by unscrewing fittings and looking. After draining the tank of course.
If you have good flow out the pressure release valve I would unscrew the plumbing from the hot out and check out the check valve.
As for the mixing of hot/cold water, you may have another issue going on. Check your shower to make sure the valves are closed (valves may be open with the button on the shower head turned off). Both the inside shower and the outside "shower". If these valves are left open it allows the mixing of hot and cold.