History:
25 years new build and private housing
Answers:
Is the breaker in your fuseboard tripping when you turn on the shower.
the circuit the lights are working on shouldn't have anything to do with the shower circuit only thing i can see happening here is your rcd is tripping indicating you have a faulty shower unit
Best to get 3 quotes
low £400
high £600
If you are re wiring the sockets you required to have rcd protection.
if you are just changing the socket fronts u don't but i strongly recommend u have rcd protection.
if the fuseboard needs changing it will have to be a metal one
It would be more efficient to have a room thermostat you can get a wireless one fitted if the wire is no longer available
I don't think it really matters as long as its in a safe place where it won't get damaged