How do I take a screenshot while the game? Is there a way of doing this without using a 3rd party utility?
You can certainly take a screenshot without using a 3rd-party program: up until recently, I played exclusively on a PC-laptop (WinXP), and used the 'PrtScrn' button -- on my laptop-keyboard, it's on the top line, just to the right of the F-keys (the positioning is similar on a standard PC-keyboard).
However, once you've got your screengrab, it will just sit in your clipboard unless you put it somewhere. Assuming that you're using Windows, you should have MSPaint installed somewhere (if you know where to find it!): open it up and just use Ctrl-V to paste the screenshot there. (I find MSPaint really clunky, so I
use Paint.NET instead). The only drawback to using 'PrtScrn' is that I use dual 1280x800 monitors, positioned top-right-corner adjacent to bottom-left corner (to minimise the cursor-crossover zone), so the entire area (2560x1600) of both monitors gets grabbed, including the blank offsets. This means that the first thing I have to do as soon as I've pasted the shot into Paint.NET, is to crop away >75% of it. (I should also clear my clipboard to free up memeory, but I usually don't).
On Macs, you can use Apple-SHIFT-3 (will take a shot of the whole screen) or Apple-SHIFT-4 (will allow you to click-and-drag to select the grabbed area). Or you can simply open up 'Preview', go to the 'File' menu, and select 'Take screenshot>From Selection' ('From Window' might also work, if you have Civ3 running inside a window -- but a Mac-player would be better able to answer that). Preview allows basic annotation, so actually works pretty well for making basic dotmaps.
On the subject of dotmaps, I just noticed the other day that the Map function of CAII
also has an 'Save/Export Map/View' function (in the same pop-up menu where you found the map-Properties). So you can actually prepare a dotmap in CAII (in the Map view, select a tile(s), hit Space to plant a 'New City' marker(s)), then either save the picture direct to the desktop (or wherever), or copy what you've done from CAII to your preferred graphics program, for further editing.