Why does hot air from my fan heater stop after 4 to 7 seconds?
#2 is not a thermostat.  It is a thermal protection safety device, and the usual culprit in cheap Cheese heaters.  I can trip if the heater is failing and the protective device is behaving correctly, or the device has failed. 

I would test it by wiring any 240V lamp, such as a small neon light(available as a component) between its terminals (parallel with it). It will act as a fault light, in normal conditions it should not light.

Don't even think of running without it.  It is an important safety device.  However you may be able to get markings off it and find a replacement on electronic component web sites in the neighborhood of 1-2 quid plus shipping.     

This is a cheap heater and it should be expected to fail after a year or two.  If you can't afford cheap, buy quality.