-Is x86_64 really any better than x86?
... yes and no. 64-bit can handle bigger numbers and can address more ram. But this can slow down "ordinary" uses.
-What kind of support doest x86_64 have compared to x86?
... the same as any open source project. What do you mean by "support"?
-Can I run programs made for x86 on x86_64?
... much the same as for 32-bit. However, some software does not have a 64bit version (firefox?) and a great body of code is written against 32-bit libraries. There are ways around all this though.
In general - server, or high-eni work is suggested for 64-bit platforms. Use a 32-bit distr for a home desktop. It is unlikely you'll do anything where you'll notice the difference.