You don't need a cage lamp. Granted cages lamps have UL ratings for CHEs but that means almost nothing outside of some company paid to get the rating. A dome will work just fine provided you have one of the required elements.. a ceramic socket. You have to have that. I have caged lamps simply because i don't like domes so it is a personal choice from taste over the years.

You can place it on top of the screen or get a pole to hang it from over the screen.

You don't need a dimmer and thermostat. The benefit of a dimmer is you can have the CHE on constantly as opposed to on/off cycling of most thermostats. Either will do and you can use both but they are unnecessary in reality.