Windows 8 is an upgrade from Windows 7 system-wise. 8 has better memory management, faster turn-on/shutdown speed, better network management and a bunch of tweaks that were not possible in Windows 7 that benefits the systems overall. The difference is not that huge, but it is there. Modern video games will most likely run slightly better on Windows 8 than on Windows 7. It will be more or less noticeable with "weaker" systems. That's the theory behind it. In practice, every computer is different, so the results will differ from a computer to another.
That said, Windows 8.1 is like a service-pack (with a fancier name and more drastic changes). It brings major updates to Windows 8 features, notably the Windows search, more customization for the App Menu and the return of a start button (but not the start menu, which Microsoft announced will be available for Windows 8 around 2015 for the next major update). If you have Windows 8, going for 8.1 is strongly recommended.
I used Windows 8.1 preview version, which didn't affect my applications/files, but going from 8.1 preview to 8.1 release forced me to re-install Windows (thus losing all application installed, yet keeping the files). I am not sure of the effects of going from 8 to 8.1 release.
For my own system and FFXIV:ARR, I can say it worked as well in Windows 8 and Windows 8.1.
For the App Menu popping during gameplay, my guess is that you press the Windows Key (the key in between Ctrl and Alt), which is a shortcut to the App Menu. That or if you still have Windows 8, if you click on the lower left corner of the monitor, it triggers the App Menu as if you had clicked there. With Windows 8.1, you no longer have the lower corner issue, since the start button is back. I can grantee though that Windows 8 and Windows 8.1 do not go into the App Menu by themselves without user interaction. It is highly probable that you do something "wrong" while playing, which trigger the App Menu shortcut.
Final words for the players out there: an operating system has two major components:
1. The user interface (what you see). That's about less than 1% of the OS (and that number is lower for most folks out there since they don't use, therefore see everything an OS has to offer).
2. The actual operating system (what you don't see). That's the remaining 99%.
If Microsoft made a mistake with Windows 8's user interface, as an operating system, it worked quite well and is definitely an upgrade from Windows 7. I would not recommend upgrading from Windows 7 to Windows 8 as Windows 7 is already a pretty solid operating system and you won't gain much outside turning your computer on and off faster. If you are good with computers though, you can easily change the user interface of Windows 8 to one so close to Windows 7, most people would never notice.
I like to think one who knows how to deal with computers, knows how to adapt the computer to him/herself instead of adapting him/herself to the computer.