You can get short heat cables in most building supply and hardware stores. You don't need to buy hundreds of feet of it.
I have also seen it sold as heating for oil sumps on stationary machinery and for battery warmers.
Anyway it was just a thought.

So if you are going to put a heater in the shed then why not just put an old house baseboard heater in there instead of rigging a light and thermostat controller?
I know baseboard heaters can be found fairly cheaply complete with thermostats. I think they are usually regulated so they don't get hot enough to cause a fire, but you might want to check and be sure.

I don't like the stand alone electric space heaters because they usually go too high in temperature (glowing red) on the elements and can start fires.

In the mean time leave a tap running just a little bit.
If there is a trickle of water flowing it is usually enough to prevent pipes from freezing.