Bad shed?

[tttaylorrr]

No I have not got around to buying one either.

unplug the UTH immediately. they are dangerous when unregulated. ALL sources of heat MUST be regulated.

I also don't know how a lot of them work either, if they would share the same outlet?

UTH plugs into 'stat, 'stat to wall, done. they're quite literally plug and play.

I really am against filling a bunch of outlets for fear of the circuit blowing.

thermostat is the only thing plugging into the wall (UTH plugs into 'stat), getting rid of the lamp frees up a spot.
regulated devices don't really pull that much power, unregulated ones run constantly and are more of a risk in many more ways than one.