I believe there shouldn't be a problem having the touchpad and mouse active at the same time. I've had 2 Logitech mice and the touchpad on at the same time and they all worked fine. Oh, one thing to note: I didn't install any of the mouse software. It may (not sure) conflict with the touchpad software. Windows should be able to detect the mouse on its own.
Unless you are putting the computer into hibernation mode (most likely by closing the lid) and then trying to use the mouse, then I don't think we have the same problem. However, once you get this one fixed, you may run into the problem I have. So here's the solution to that one:
The problem I had was very specific: When the computer hibernated, it shut down power to the USB ports to conserve energy. Sometimes when you bring the computer out of hibernation, it fails to wake up the USB ports. I could restart the computer and the ports will work fine, because I was starting from scratch. Once I hibernated, though, that was it for the mouse. The solution (so far as I know) is to not allow the computer to turn off the USB ports.
This would be to right-click on "My Computer". Go to "Properties". Select the "Hardware" tab. Then select "Device Manager". Expand the "Universal Serial Bus controllers" and you should see a few lines with "USB Root Hub" Double-click on those, select "Power Management" and then see if the "Allow the computer to turn off this device to save power." It should be unchecked.
Ok, back to the original problem:
Assuming you have rebooted fresh and the mouse doesn't work. Did you install the drivers for the mouse? If so, I would recommend uninstalling them as they may be conflicting with the touchpad drivers. Have you had to wipe the machine and re-install Windows for a fresh start? (Not that I'm recommending it, but just making sure you have the Dell's pre-loaded factory defaults.)
If you have one, try plugging in a non-wireless mouse, just to make sure it isn't the wireless connection. If that doesn't work, then try a generic USB device to see if you can get a "New Hardware Found" bubble to appear in the taskbar. If you plug in various USB things and the "New Hardware Found" isn't coming up, then we have to focus the trouble shooting on the ports.
Another alternative is to try your wireless mouse on another computer. As of this point, we're trying to isolate where the problem is. (Mouse, Windows setup, or hardware.)
Sorry to be long winded, but I hope it helps.