So, all I want to know is if I put a not self-priming pump in my sump, and when the power goes out, then comes back on, will the pump still pump water? Or burn up or what?
A submersible not self-priming sump pump will be fine as long as it stays under water. A self-priming pump is a fiction; they are pumps that create a strong suction. They hardly ever live up to the title self-priming and you have to add water to the line to get the siphon going. This said they are better pumps because they tend to be more powerful and better at moving water.
As long as the volute is flooded with water should start. I'v never had any issues with my external pump starting. The only time I did was when I sheared impellor teeth off after my heater went through it.