A under tank heater is not meant to heat the air in a cage. It is meant to create a hotspot for the snake to lay on and needs to be controlled by a tstat.
As for sticking it to the bottom, I wouldn't. Use foil and stick it to the sticky part of the mat. Then use aluminum foil duct tape to secure the uth to the cage bottom. This will more easily allow you to remove the uth if needed.
As for the size of uth it is usually suggested that the heat mat cover no more than 1/3 of the cage floor.