Every time I pause or unpause Netflix on Apple TV it starts playing iTunes music on my computer, even with wi-fi and home sharing turned off.
How do I disable iTunes from launching when using Apple TV?
It's probably due to the fact that your remote is detected by both machines at the same time. They obviously do not know to whom the command is addressed, therefore this happens.
One way to stop this is to pair a remote with one of them, but not both. On your Mac mini, go to System Preferences and then Security. In the first tab (General) you will see a "Disable infrared receptor" and a "pair" button.
On the Apple TV, go to Settings > general > remotes from the AppleTV main menu, and then choose "pair Apple Remote".
You can pair 2 different remotes with each machine, but you won't be able to pair one remote with both machines.
Hope this helps!