First things first - HA is a headless server. Connecting a screen to it will only show you a command line - you won’t see a dashboard or anything visually pleasing. You will just see white text on a black background.
There are ways you can show a dashboard on a directly connected screen (like this one), but it’s not really a beginner project.
There is also this guide specifically for your Crestron panel, but I believe that is just for using it as a dashboard - the actual HA server is running somewhere over the network on separate hardware and is not directly connected to the Pi.
Regarding your PoE question. If you want to know whether you can use the Pi’s ethernet port as as a power source for your Crestron panel, no, you can’t.
Use a PoE switch or power your panel with DC voltage (or whatever type of voltage it accepts).