Should you run the pump if the chlorine level is too high?

Answer 1

That depnds on what you mean by "too high". Your chlorine shouldn't be higher than a regular shock, about 10-12ppm, unless you have a really bad green algae bloom that required chlorine up to 25ppm.

Even at 15-20ppm for chlorine your pool pump should be fine.

Answer 2

Yes it will mix it in better however it is not a must.