Window 7 home will be fine for gaming. Pro and ultimate gives you the ability to join a domain, on top of that ultimate gives you multiple languages.
your ram limits will be
Version
Limit in 32-bit /64-bit Windows
Windows 7 Ultimate 4 GB / 192 GB
Windows 7 Enterprise 4 GB / 192 GB
Windows 7 Professional 4 GB / 192 GB
Windows 7 Home Premium 4 GB / 16 GB
Windows 7 Home Basic 4 GB / 8 GB
Windows 7 Starter 2 GB / 2 GB