IF you where using your day/night setup for a source of heat then no you can leave the CHE on 24/7 connected to a dimmer and adjust accordingly for nighttime drops and such. Light during the day is not really necessary either, they do just fine with the natural light of your room.
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.
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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 a zoo med dual dome. It says it has ceramic sockets but says not for use with a CHE is there a brand that is best?