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.

Down here in Florida people do two things when we know a freeze is coming. Wells down here have spigots on them that you can open and let the water trickle out at enough of a rate that the switch comes on often enough to avoid freezing. Another trick people have done is taken bird houses, put them over the pressure switch (That's what we call it here, at least) and install a light inside the bird house. One you can plug in or wire to a switch and flip on. The heat of the incandescent bulb keeps it warm.