Questions and Answers for GoControl Z-Wave Plug-In Remote On/Off Appliance Module

It should work just fine if you are within 30 feet or so. Z wave / zig bee transmissions go through walls and doors fairly well.

That is the purpose so you can contol it from anywhere. The cavat is how far your z-wave hub is away from it which is what you need to control this. ie smartthings hub or wink hub. But in must homes that shouldn't be a problem

If the oil heater can be turned on simply by plugging it in then this unit can control it.

Yes.

Yes, you can locally control this unit.