Eyefinity is mainly being marketed at gamers right now, and the big draw is multi-monitor gaming. It will allow you to use 3-6 monitors combined with your game split across all displays. It can do what you are asking, but then again so can any non-eyefinity set up. Eyefinity is designed for 3-6 displays.
What you want to do is just set up a regular old multi-monitor set up, either using the windows vista/7 built in monitor set up or through software such as Ultramon.
But you will not have fluid movement between the displays. When you have a game active, that game locks your mouse to the parameters of the game and the only way to release it is to use a hot key set up, or through something like alt-tab. Doing this means you will lose control of the game while you use your mouse/keyboard on the other screen for what ever you need to do, and then you will give control back to your game. The other screen will update when ever your messenger or what ever you have on there gets some action, but you will still have to lose control of your game to go over and do anything about it.
Eyefinity is only support on Windows Vista and Windows 7, XP is not supported. From my understanding this is just a driver issue, but the reality is XP is dead as an enthusiast platform so why would ATI want to waste time supporting it. XP is not very good with multicore processors, has no TRIM support for SSDs, will not support anything beyond DX9, and beyond that...Microsoft will axe support for XP before long anyway. So don't expect any XP support from ATI in the near future.
Honestly if you want to have an active pc experience on that 22" LCD while you are gaming on the TV..just go buy one of the cheapo Atom boxes and have that as your 2nd pc. A $200 box such as
this will easily run your other display for surfing the web, chatting, etc while you game. It's a much better route in my opinion, than dealing with more software, overlays, windowed gaming, and all that other hassle that a lot of games crash as a result of.