I suppose it depends on which 64 bit OS you are talking about. For example with Windows, the 64 bit version of XP never really got off the ground and was poorly supported, but Vista and Windows 7 64 bit are perfectly stable. I've used both and my current Windows 7 Ultimate 64 bit has not caused me any problems.
And yes, 32 bit software works fine with 64 bit Windows.
You shouldn't have any problems with Gimp although I see from http://www.gimp.org/windows/ that they do not mention Windows 7, nor 64 bit. However they do mention Vista.
If you cannot get Gimp to install on Windows 7, try the Windows 7 Compatibility option which is included and designed to "make older programs run in this version of Windows".
I have read several articles on this and some say the 64 bit is unstable and not to use it. Same article said to use Gimp 32 bit (can you do this on 64 bit computer?) I have been using Gimp for years and really like it. Use it on my XP Notebook and would like to continue. Is anyone using Gimp with Win 7. Which version are you using.