thermosiphon actually works pretty good when all parts function as designed. Some older gasoline tractors and diesel Kubota's used that priciple as they didn't even have a water pump or thermostat. Old Kubota diesels had a whistle that would start squealing if the temp got too high (envision an old tea pot).
Most engines will heat up a little hotter after shutdown as theres no air moving through the radiator for heat transfer. So it's not unusual to see the temps climb after shutdown, This is especially visable on mechanical temp gauges.
re. idle cool down. I always like to idle most all my engines down for cool off when I can. I always do this with my diesels.