Pretty much what's been said above. I'm going to assume you're upgrading from WinXP here.
There was a huge jump from Windows 98 to Windows XP in terms of backwards compatibility in that there really was shit-loads of games from the 1990s that will not work on XP, and consequently will usually not work on Win7 either. The differences between XP and 7 are less pronounced. Out of around 600 games, I've found ten that I needed to boot up my network-isolated XP partition for.
All the GOG games should be fine.
Home Premium is more than up to the job. If all you're using the machine for is gaming and simple home office stuff, there's no need for Pro. You should bear in mind though that older disc games may struggle with the 64-bit side of things, as many 90s games using InstallShield used a 16-bit stub that will not launch on a 64-bit machine. For some games, people have created customised installers for them, but don't necessarily count on it.