Interstellar space behaves as an almost perfect blackbody radiator, at a temperature of 2.725 Kelvin, due to the cosmic microwave background. As such, it does "emit heat", albeit an extremely small, but measurable amount.

Probably not very well if the drone is the typical (quad)copter type since the shield will likely obstruct the airflow from the propellers reducing lift. The added weight of the shield is also likely to be a problem in small drones as most are already weight constrained. Although in small electric copters like those used in Ukraine, I'm not sure how often thermal imaging is used as a primary detection vector (due to size, distance and daytime thermal delta from below).

Thermal is definitely a primary vector in detecting larger drones like the Reaper due to large combustion or jet engines, but in that case the challenge isn't cloaking the drone, it's cloaking the heat of the vented exhaust that's likely leaving a thermal 'contrail' across the sky.