If some doors work & some don't, I would check the wiring into the door. Almost all the similar faults I have seen are caused by the wires snapping inside the rubber sleeve.
Easiest way to check for continuity is to remove the door card and see if there is a feed to the solenoid.
If they work, but there is no locking function, check for mechanical function, such as broken clips or disconnected levers.
As for the heater fault, I would guess either an airlock, partial blockage or partial function of the control valve.
However, my first thought would be a duff thermostat. They usually fail safe, in the fully open position. Remove it and pp it into a basin of freshly boiled water to see if you can see it open. If it's already open, replace it with a new one.