Should my sump pump be running when the water level is below the pump?
Get rid of your mechanical switch.  

Go with a HC6000v2, it has two sensors that you place at the activation and de-activation point and they are easy to adjust.  

To your question, as long as the pump can pull water and is not sucking air you are good.  I typically try to have the pump operate for as long as possible as the pumps are rated for so many activations so preventing it from going on as often will extend the life of the pump.

https://www.amazon.com/HC6000-Hi-Lo-Controller-Float-Switch/dp/B006AU4L4U