The depth is about 4 1/2 " very thin.
Yes, it gets hot enough, on the metal front, to burn you if you touch it while it's heating.
Should be able to use it all night in a 120 sf greenhouse since it is rated at heating up to 160 square feet. It has a built in thermostat. Regarding your question about attaching an external thermostat...In the manual it states: "In order to prevent hazards due to the inadvertent resetting of the thermal cutout, this appliance must not be supplied through an external switching device, such as a timer, programmer, separate remote-control system or connected to a circuit that is regularly switched on and off by the utility." So it appears, based upon the manual, that you must not attach it to an external thermostat. It would be helpful if the manufacturer could comment and elaborate on the external thermostat issue.