Should my refrigerator run every time I open the door?

No, only when you're chasing it.

Seriously though, yes, especially if it's an older model but it depends on how full it is, how long you keep the door open, what your air temperature/humidity is, etc.

New refrigerators have compressors that are designed to run 80-90% of the time which makes them more energy efficient. Cycling the compressor on (then off then on then off) takes more energy than running the compressor steadily.

Foods, liquids, ice "store" the cold - the more filled the box, the less it is warmed-up when you open the door.

It's pretty typical behavior. The thermometer notices that the inside temperature of the refrigerator has gone up, and it will run the compressor to cool things back down.

Every time...no. Lots of the time...yes.

If you keep it open too long it can, to keep it at the proper temperature. But not necessariy every single time. If you can check the internal temperature to be sure its cooling enough to keep foods fresh.