Yes it will! While Win7 Requires that you have at least 2GB, it is not all used by the OS. That is perfect for gaming, however the more the merrier! If you use 32 bit OS, your RAM will be used by the OS and is not good for gaming. But for you, that is just fine
Yeah, you'll be fine. I used to run my copy of Windows 7 64 bit professional with 4GB of DDR3 1333MHz RAM. The system ran fine, and there was usually enough RAM for gaming, very few games can use more than 2GB of RAM, so with 4GB you should be fine unless you are running lots of stuff in the background while you're playing.
That said, RAM is dirt cheap right now, and if you want to be a little more futureproof you may want to invest in 8GB now. That way your system will be ready as more and more applications become 64 bit capable, and thus are capable of using more RAM.