so i've downloaded a copy and installed it onto my cousin's computer without registering....which im guessing is semi-illegal
Not having registered would indicate that you've downloaded a pirate copy from a third party. And yes, that's illegal. If you register the game you can download it legally through SDC, though you're not allowed to share the game with other people (if you're giving it to your cousins, you'd need to get rid of it yourself, or buy a second copy).
Anyway, first off, you seem to be confusing registration and activation. Registering is a one-time event where you link your serial key to your account so you can get patches, or download the whole game via SDC. Activation is done each time you put a patched copy of the game on a new machine, so it can be played (though if you install via SDC, it's done automatically for you). Unpatched retail copies do not require activation or registration to play, but if you do want to patch, you'll need to do both.
Registration and activation both require your serial key; unless you get GC2 Gold, DL and DA will use different keys.
The first thing you'd need to do, if you haven't already, is register GC2. This allows you to get updates or download the whole game. To do that:
1) Get
Stardock Central (our download manager, used to get patches or even the entire game should you not have your discs handy to install from).
2) Install it and log in with the same account you made for the forum.
3) Go to the Tools menu in SDC, and choose Register Products.
4) Pick Games on the left in the Register Products window, then find GalCiv2 in the Product pull-down menu. Enter your email address (same as forum account, again) and serial number, then click Register.
Once that is done, you'll be able to download, patch, and archive (as above) GC2 via SDC. Now, to get DA, you can buy it directly from the Purchase page on the site here if you have a card compatible with the major US credit companies, or if you have a verified paypal account. In that case, just proceed through the store, making sure to give it the same email address as you registered GC2 with when it asks for one.
Once the purchase is completed, DA should be automatically added to your account (if not, you should be provided a serial key for DA by email--you can register it by the same process as above, just pick Dark Avatar in the product list rather than GC2).
Once you've got that done, you'll be able to download both GC2 and DA from your SDC account, and archive them for manual transport elsewhere.
If that's still confusing, there's a simple option. You appear to be in Australia, so you should be able to find GalCiv2 Gold (published there by Auran) in a local game store. GC2 Gold includes both DL and DA, and as above the unpatched retail version doesn't require doing anything with the serial to play. So you could simply give it to your cousins, install it on their machine as any other retail PC game, and they can just play it unpatched until such a time as you're ready to deal with SDC again (or if a standalone 'public' patch is released after the final DA patch).