Not clear from the Tesco description, but this fire has to have a heating element and a fan to blow the heat out. That's what cause a any noise.
Frankly can't see why attempting to deaden any sound would make a difference, and fires in general have to be mounted on non-combustible surface.
Yes, electric fan heaters have a thermostat that switches off the element once at temperatures.
Are you really sure you want a thing like this?
Not good value for money imho.