Yes and no. Of course, the save games are not on the CD. However, they are on your hard drive. Assuming you did the default install, they'd be in C:\Program Files\Infogrames Interactive\Civilization III\Saves for the vanilla game, and \PTW\Saves and \Conquests\Saves for PTW or Conquests.
If you have complete, I'm not sure on the directory path for the saves, but it should be easy to find.
You just need to copy those to a flash drive or some such, and move them to the new computer.
And for the editor you'll need at least 1024x768, and it would be better to have 1280x1024, because the Custom Rules screen is just slightly bigger than 1024x768. You can deal with it, by moving the dialog box for the Custom Rules so that the title bar is as far up as you can get it.