Does the dome thermometer indicate a different temperature than the thermometer at the grill/grate?  


This is not surprising. The grate/the grill is closer to the fire. Therefore, it is regularly the case that the temperature measured at the grate is much higher. A difference of, say, 35 degrees is certainly not unusual.  


Using a plate setter can also affect the temperature at the grate and in the dome. Does the hot air rise directly, or does it go along the stone? That too can cause temperature differences in different places in your kamado.  


Tip: want to achieve a high temperature for e.g. baking pizza? Then keep in mind that when your dome thermometer shows the desired temperature, the pizza stone has already reached a much higher temperature. So take this into account in determining the ideal temperature. 

So what is the ideal temperature for baking a pizza? There is no single answer for that, everyone has their own preferences.