You don't say what app you're using to play your music. Suggestions might change depending on what you're doing.
not sure how to fix that problem with the aTV. Rebooting your router and other network stuff might help, no guarantees, but it shouldn't hurt anything.
there are a couple options that might help or make it less annoying.
1. a lot of TVs have an option to turn off the screen while still keeping everything else powered on
on my LG it's in picture settings -> energy saver.
I also have the option to add screen off to the "quick settings" menu that pops up first when I hit the settings button on the remote. so I don't have to dig all the way into settings.
2. another thing that might make it better for you, is to enable screen saver when playing music, it's still pops up a little notification in the corner when the song changes, but maybe it'll be less annoying.
That setting is in the screen saver settings on the aTV.
3. Look into Plex, but this won't work if you're playing DRM music,
Just point it at your music folders and it will do the rest. You'll have to to select music from the aTV though
It also works great for video, and will categorize those.
With their premium upgrade, you can also sync music to your mobile devices if you want.
Or as long as the server is on at home, you can stream to pretty much any device from anywhere with internet. (no subscibtion needed for that)
This is assuming you're playing files you have locally, if you're streaming, get the app for that service on the aTV.