Stay safe working with portable nuclear gauges 

Just use a term alara, which means as low as reasonably achievable. So safe distance when testing, keep stored in the truck, close rod in the gauge before recording test results, don't open the case, use a mirror to check the shutter and when servicing

None, just a dosimeter but even that isn’t necessarily needed. With proper alara technique there is literally no threat

A dosimeter, preferably the electronic alarming type. You won't get much exposure from these in normal operation. ❤

cesium 137 for gammas (density) and americium-beryllium for neutrons (moisture content)

No that isn't true. The sources are 15mCi Cs137 and 40mCi of AmBe for density and moisture. Both sources are well shielded when the device is not in use and simply stepping back 10 feet or so while it does its thing gives negligible exposure to radiation. If you have concerns you can purchase an alarming dosimeter.