Should my sump pump be running when the water level is below the pump?

The obvious response that I should get out of the way here is, find and check your manual to be sure;

I believe the opening for the sump pump has to be submerged. I don't think there's an issue if the inlet is submerged but the top of the pump isn't, for example.

However if the water level isn't up to the opening for your pump, and it's not able to draw water in, the pump should not be running. Air getting into your pump while the pump running can contribute to the pump burning itself out.

To me it sounds like the float switch is stuck in the on position and the pump is overheating and shutting down. Almost all pumps require water to cool and lubricate the shaft seal. Some pumps can be damaged in as little as 30 seconds without proper water levels. So I would actuate the float up and down (move the switch or wiggle it to see if it is stuck) this may free it up but if it keeps cycling on and off with no water it is probably overheating and the internal thermal overload is trying to keep the pump from melting down.

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