For Windows 2000...
Even in an extended desktop the task bar only lives in one monitor. You can choose which monitor it lives in but it doesn't extend to the other monitors.
As for windows opening up in both monitors, closing them and reopening them again and having them open in the monitor they were last running in ... that capability seems to be dependent on the software you are using so long as you've got your system set up ok. Most of my windows will remember where they were when they are closed and when I restart the application they open in that monitor again. Windows Internet Explorer seems to be very consistent with this and would be a good test to see if your system is set up ok. Open IE it in one monitor, move the window to either monitor and then close the window. When you start IE again it should open right where it was last closed. Try it in both monitors. If that works then your system is probably fine and you need to look other places for your problem. Specifically - call the vendor of the application that isn't behaving for their help.
RealTick had (and may still have) a bug where the 'primary' monitor must be assigned as the monitor to the furthest position to the right (and if you have multiple rows of monitors, the upper row). Most people are able to get RealTick to cover all monitors without doing the workaround I'm about to mention but the drawing tools don't always work in all monitors if you don't have the primary monitor assigned to be the upper leftmost monitor. For anyone that discovers they have this problem you can do the following: First tell windows 2000 that the primary monitor is your upper leftmost monitor (Display Properties in the Display control panel). Once you've done that you restart RealTick and shrink it to fit one monitor, move the window to the primary monitor to it's furthest upper left position and close RealTick. Finally you open RealTick and stretch the lower right corner of the window across the monitors and close RealTick. Now every time RealTick starts it covers all monitors. As long as you don't rearrange your monitors often you shouldn't have to ever do this again!
I have some other less capable shareware tools (and actually a commercial and popular set of virus protection / system tools) that are very good at some tasks but their multi-monitor windowing code isn't very smart. They continue to open up in the primary monitor all the time or just plan work best there. This is rare however. Most of my applications I've been able to get to open where I want them and for them to remember to reopen there.
There are shareware and commercial software packages to help force applications and windows to 'do the right thing' about opening where you want them to. I use a commercial package that came with my video cards that helps out in this regard. It's HydraVision - Appian's Desktop Manager. I think they sell it separately but there are various solution providers for such things. I can't vouch for how well that would work for you or not.
www.appian.com or use a search engine to look up 'multi monitor desktop manger' or similar for other options.
All my experiences are based on Windows 2000.
Hope this helps. Good luck!
- David