Xbox One controller WILL NOT work with the 360. I have both consoles and have tested. Microsoft stated that the controller shall not work on 360. Meaning Xbox One controller works only with Xbox One and 360 controller works with/only 360 console.
But PS4 Dualshock controller works on 360 not sure about Xbox One though.
Cited
This entirely depends on the game.
For most Xbox Live purchases, the answer is no. Since the Xbox 360 and Xbox One have a completely different architecture, binaries are not compatible between them. For a game to be released on the Xbox 360 to be released on the Xbox One, it must be specifically released for the Xbox One. Due to this, many games available on the Xbox 360 Marketplace simply are not available for the Xbox One.
Some games offer a free upgrade. For example, Call of Duty: Advanced Warfare allows people who purchased the Xbox 360 version of the game to upgrade to the Xbox One version for free. The same goes for PS3 players, they will get the PS4 upgrade free of charge.
Other games offer an upgrade for a reduced price. An example would be Minecraft:
Owners of Minecraft: Xbox 360 Edition can upgrade to Minecraft: Xbox One Edition with a substantial discount. All downloadable content previously purchased for the Xbox 360 Edition is available free for redownload on Xbox One.
Finally, some games you simply must purchase again. Plenty of games have been developed towards both platforms, and have released for both the Xbox 360 and Xbox One. These games you would have to purchase again if you wanted to "upgrade" to the Xbox One.
The games you mentioned (Walking Dead 1 & 2 and Wolf Among Us) fall under this category. While the games have been re-released for the Xbox One, it is a full purchase. There is currently no free or discounted upgrade program for these games.
Best Answer
There are a few answers to your questions, so i'll try to be brief. The short answer is no, not officially. But, due to the improved hardware on your Xbox One, titles from the Xbox 360 stick to their frame rate caps a lot better than they did on the 360. For example, GTA V might've given you 22-30 frames on the 360, whereas the Xbox One will more than likely give you the full 30 at all times. Also, a lot of TV's upscale, so that could help with the visuals. Other than that, The Xbox One X is the only (Microsoft) console that actually Performance Mode for certain titles.