How to extend the desktop with an Nvidia 8800 GT graphics card

The first place to check is Nvidia's System Requirements for 3D Vision. Without using SLI (which would require two or more of the same cards), you can't do what you're asking. This is just due to the way it works - rendering such a large image, and then sending it to another discrete video card before displaying it would be such a performance hit it wouldn't be worth it.

While this uses the 3D Vision technology, you don't have to run games in 3D mode - you can run them in 2D mode as well. In this mode, you also don't have three desktops, but rather one triple-wide desktop (at the desktop level). See this thread for more details.

Do note that the game must also support the 3D Vision Surround Technology. Note that using three monitors on the desktop level is a completely different story (if you want discrete desktops as opposed to a triple-wide one), and you can see details for this below.

Additional information: As you already know, you need to use an additional video card to get the triple monitor setup working. Additionally, if you wanted to use SLI/Crossfire in this setup, you would have to check Nvidia's website to see if triple monitors are supported (e.g. I had 2 x 8800 GT's in SLI, and had to disable SLI to get triple monitors to work - this does work with some newer Nvidia cards however). See this page for more details.

Using two video cards in non-SLI is a completely different story, and quite possible with any combination of cards. My recommendation is to use your one center monitor as your "main" monitor (and have the 560 be that monitor's hardware acceleration), and plug the other two monitors into the 260. That way, 3D applications that output to your main display won't take a performance hit (those running on the additional monitors will - you might want to put one of them back on the main 560 if you need something else hardware accelerated).

The core requirement from Nvidia's website to use an additional video card to drive an additional monitor is:

Additional monitors (up to 6 monitors total enabled) may be enabled by using either a motherboard GPU and/or a PhysX capable graphics card (GeForce 8 series or higher with at least 256MB of memory) that does not have the same GPU as those that are SLI enabled. More information regarding multi-monitor in SLI can be found here.

Long story short: With the hardware you have, it's impossible. If you got another GTX 560, then it would be possible.

(Posting comment as answer, per OP's request)

There are plenty of applications (DisplayFusion being the best that comes to mind) that will give you duplicate taskbars, which is way better than doing this old-fashioned spanning because windows behave properly - you can still maximize to just one monitor, etc. There are better ways to accomplish multi-monitor workflow than spanning these days, thank goodness!

That second behavior is not something I've seen - I'm guessing it's a consequence of that particular graphics driver and/or chipset disabling the secondary output while the machine is locked (at which point Windows will shuffle everything back onto the primary). Most systems don't do this, but I've not tried these exact graphics, so I'm just conjecturing. If that's the case there's probably no good workaround.