You don't need a jailbreak. I do it all the time. All you need to do is mirror ipad screen to ATV. it comes out a bit smaller but works fine.
Better not. Anything involving wireless AirPlay screen mirroring is awful, both resolution-, framerate- and compression artifacts-wise, compared to native playback.
Your best choice is, as I've pointed out above, Infuse, if you want to play back non-iOS-native files natively over AirPlay. And it works even w/o JB'ing.
(BTW, I've published tons of articles & comments on this. See for example
https://forums.macrumors.com/threads/1680468/ )