We have a similar setup at our school, approx. 1300 students and an Apple TV in every room, most of which are connected to Dell projectors and Promethean interactive whiteboards. Like your setup, our ATVs connect via Ethernet.
And yes, we've had our fair share of teething troubles since adopting Apple equipment in 2012, particularly with dropped connections. Apart from poor IT practise and usage from staff we've had a few network issues that, once identified, have made things much more reliable.
Before last summer all our access points were set to dual-band operation, allowing the Macs to select network settings automatically. This worked reasonably well until a Cisco firmware update in the summer when things went downhill. Airport diagnostics indicated the 2.4GHz band was pretty busy so we made the Mac VLAN 5GHz-only. Having all clients connecting on 5GHz has massively reduced the number of dropped connections and reliability overall has significantly improved. Incidentally this have the knock-on effect of killing AirPrint on our Epson Workforce printers so they've had to be wired in.
Our biggest problem though has come from teaching staff using MS PowerPoint. Prior to 2012 all our staff used Windows and Office for all their needs, and it was deemed necessary to ease the transition to the Apple environment by giving Staff access to Office Mac 2011. I'm no fan of Office but it's what the staff are used to. Our PowerPoint problems come from staff thinking they are saving time by loading up a day's worth of teaching material into PowerPoint, which then frequently crashes and takes AirPlay down with it. Occasionally we can sort AirPlay issues out by cycling the Wi-fi off and on again, but frequently it's more of a case of logging the teacher off and back in again, or restarting OS X in the worst cases. Teachers using Keynote are almost never affected.
A couple of other things...
Occasionally (and generally while using PowerPoint) a teacher will try to connect to the ATV in their room but it will randomly connect to a completely different ATV elsewhere in the building. We nailed that by passwording each ATV using the room number as the password.
We've also had issues recently with a number of ATVs refusing to wake up automatically when attempting to connect to them. This has come as a result of a software update which failed on those units. They would otherwise work ok but would have to be turned on manually. Restoring the units via Apple Configurator brought them back to full functionality. - Note, this can also be done via iTunes but iTunes doesn't cache the downloaded firmware and so does a fresh download for each unit to be restored. Restoring via Configurator is much quicker as it checks to see if it has already downloaded the firmware.
Hope this gives you some ideas...