You are using the most complicated and failure prone options. Streaming video over WifI rarely works well. In your case you are trying to do it twice concurrently. One WiFi connection to the device with streaming video and then a second wireless connection to AppleTV. On top of which AirPlay is horribly bug ridden and un-reliable.
Just select the content you want to watch from the ATV menus. It works very well and with none of the issues you get with AirPlay.
And to your last question yes you will have fewer issues if you use a wired connection for ATV.