Pueblo, I'm not an expert, but I recently upgrade to an AMD Athlon 64 and had these same questions. Before I did the deed I found the answers as reported by others. So some of what follows is "hear-say", and some is first-hand reporting. I'll try to be clear which is which.
First, yes, the AMD Athlon 64 is a true 64 bit processor. It is capable of running 64 bit instructions, but can also run the 32 bit instruction set with only a small degradation in performance. That means that you can load up the 32 bit version of your OS and run it. I did just that with Mandriva 2007 Free. I hear reported that the difference in speed experienced between the 64 bit version and the 32 bit version is negligible. You won't notice.
If you were to run 64 bit applications on a 64 bit machine, yes, it would probably be faster than 32 bit applications on a 32 bit machine that is otherwise identical. The difference is reported to be not as great as the difference we all saw when upgrading in the mid '90s from 16 bit machines to 32 bit machines. Most say that it's not worth the effort to upgrade if that is your only reason, but most people upgrade to 64 bit machines for other reasons and get the marginal performance boost in the bargin.
Your last question about the applications is a good one. Currently not all applications offer 64 bit versions. There is a complication with apps. like Firefox, which do offer a 64 bit version, but then must use 64 bit versions of plug-ins, which often don't exist, yet (most notoriously, there is no 64 bit version of the adobe flash plug-in yet). Until those gaps are filled (and boy it seems to come slowly) you'll probably find that there are some things you just can't do that you use to be able to do.
The situation is improving (always improving...) but slowly.
My impression is that most are upgrading the hardware, but sticking with the 32 bit OSs and applications for now.