Q: Does AirPlay Mirroring also work with ATV connected to a LAN?

AirPlay Mirroring should also work with the Apple TV connected to your router by Ethernet without any problem, or at least, that's what other users said. To be sure it will work, use Wi-Fi, as stated by Apple > http://support.apple.com/kb/HT5209

Yes, it works perfectly fine.

My ATV is connected via Ethernet to eliminate signal, speed, and interference issues.

I am able to AirPlay from my iPhone without any issues.

Yes, as long as both devices are on the same network there is full functionality.