My MacBook Pro 17 is from 2009. I modified it a bit, by adding more RAM. Its last (ever!) new OS is El Capitan.
The update to OS X El Capitan and your Apple TV 3 will work for AirPlay, but unfortunately, you MBP is too old for either AirPlay or AirPlay Mirroring.
My iMac Retina is from 2016, and carries the latest OS, 10.13.2 High Sierra.
... and this should be just fine for AirPlay.
My first interest is in playing a movie via AppleTV to my TV (Sony Bravia), either from the MacBook's built-in player, or the external unit I have plugged into the iMac.
Eventually (i.e., when this first task is finished) I'd like to mirror either computer to the AppleTV, and maybe use a Bluetooth keyboard too.
Again, you won't be able to do either with your MBP. The iMac shouldn't have an issue with either. However, even though you can mirror your iMac's desktop to the TV (via the Apple TV), you would still need to control it via the iMac's keyboard.
Mirroring is just that, you are creating a duplicate of your Mac's screen and projecting it on the TV.