Performance wise, gaming on Windows 7 or 8.1 will largely be the same. And yes, on windows 8 there have been some issues with BF4, force quitting and error whilst joining games etc, but now most of them have been ironed out and are now non issues. If you choose Windows 7 and you're only gaming, go for Home Premium.
Windows 8 has been out for long enough to have new drivers and bug fixes for the newest games. So I still say it is just a matter of which OS you like the most. I, however just as a personal opinion, would stick with Windows 7. As for the package, as I would call it, I would stick with home premium. Till this day I have no idea why people get anything more than home premium. Listen closely, I use the term "people" to describe persons who barely know how to work computers to the persons who get it just because it is supposedly "the best of the best". Professional and Ultimate are primarily reserved to those who know of their features and put good use to those extra features. There are no added "performances" between home premium to ultimate. Only more "features"; features that you don't need for gaming.
Seen alot of infos on google saying that Windows 8 it's better than Windows 7 when it comes to BF4. BF4 was mostly optimized better for Win 8, and the difference could be seen even on youtube, but not by much.