Check this out
https://hcsonline.com/support/white-papers/recommendations-best-practices-for-wi-fi-and-apple-devices

My school has a very similar environment and experiences similar issues, have you been able to make any progress on them since your post a year ago?

I'm afraid I can't say we totally solved the issues, but we did make a lot of progress. Some things we changed:

• We replaced all of 3rd gen Apple TV's by newer models (4th and 6th gen).
• Almost all of the Apple TV's are connected to the network over ethernet.
• WiFi channelization: limited to 20.

The situation got much better, but still, sometimes on certain locations we have a problem with certain types of streaming video. Sometimes we can solve this by factory resetting the Apple TV, but not always. It's still not clear what causes these problems.

Thank you so much for that information. 

We don't experience that particular issue you're describing, but we don't use that many MacBook Airs. It's mostly iPads we are using.

Some other things to consider:

• WLAN: only use the 5GHz band and use as many bands as possible
• WLAN: use OFDM only
• AP's: limit the TX power, so that the clients and Apple TV are connected to the nearest AP
• limit the number of Apple TV's per VLAN (10–20 max)
• Apple TV: never sleep

Interesting. I've been steadily plugging away on similar issues for the last year and a half and here are some insights of my own:
Firstly, for anyone browsing and reading this without a lot of background info, airplay has a split in it's functionality. There is airplay mirroring and airstream. Most of the documentation I have found regarding to this is either old, vague or reverse engineered so based on observation these are the characteristics.

1.  Airplay mirroring will work regardless of whether or not the apple-TV has an internet connection, however it suffers when playing back content with a lot of motion in it (more noticeably when streaming the video at the same time). One sidenote here is that copy protection can trigger on mirroring similar to using a cable for an external display. 
2. Airstream can transition automatically from airplay mirroring when videos are put into fullscreen, however rather than relying on this it's better to use the airplay button in a video's playbar to trigger it directly. Airstream is capable of precaching apple video formats and MP4's when using a compatible video players such as quicktime, thus preventing stutter but causing fairly significant issues when jumping to new points in the video before the entire file is finished transferring. Notably at the present time airstream appears to only be usable with videos played in safari. 
3. Compatibility for airplaying videos streamed from the web relies nominally on three things. The apple TV should have access to the internet (I've ended up using static addresses and whitelisting the range because of environment issues), the source should be from a compatible app or video player and the service itself doesn't conflict with the protocol. 
   To elaborate on the latter here is an example, TV4Education is used by one of the sites I support, it hosts content internally and redirects from other sources as well. One of their patches broke airplay compatibility on their internally hosted videos due to authentication issues. With a service like youtube when a video is airstreamed there is a certain mechanism to pass through authentication and play it as the user of the source device. In the absence of working passthrough, a service that requires an account to access content will cause a blank screen whenever you try and airplay something. 
4. Some video hosting services will function via their dedicated apps even if their in-browser players do not support airstream. 
   
   

With that out of the way, some issues that have cropped up which I believe we determined the cause of are as follows: 

Attempting to connect to an apple TV multiple times in a given window can cause a state where the connection requests 'stack' and leave it visible but unresponsive to further connection attempts until rebooted. 

Airstream in some versions of safari doesn't work on youtube unless the user is actually logged into a google account. 

I've been experiencing trouble with detection on certain devices in certain locations and hadn't considered that it could be a VLAN issue, which in hindsight does explain a lot of the troubles I haven't been able to resolve.
Thank you for the information, I hope the observations I've written here are helpful to someone else.
Since they are based mostly on firsthand experience and inference, if someone with access to good documentation does pass by, please give it to me so I can see if I'm wrong!