Two questions:

I'm trying to launch AoW2, and the game is demanding a password. Guessing and hoping that this was not a registration number that was supposed to be supplied, I tried just entering in my own password, and that failed to launch the game.

1. Were passwords supplied by Gog, and if so, how are they accessed?

2. How can I set a download to a different drive? The C drive is getting full, but there does not appear to be any way to alter the default.

The GOG installer has a neat little button named "Options", there you can change the install path.
And you can download to whichever folder or drive you want, it depends on your browser settings.

I want the URL to the shortcut in the top right corner.

Looking at the text file on the left maybe better not ;-)

@bearcat: You should better crop your image that only the message-box is visible. I can't really help you, but I don't believe the problem belongs to the path (it seems you're running WinXP, so no problems with right-management). Do you have installed another version of AOW before? Than maybe an old registry-key is the reason for the serial-number request (just my guess).

The reward outweighs the risk. :P

Regarding the OP's problem I don't know. What I would have done in his place is to check if any image program like powerISO or something similar is messing with a old copy protection in AOW and check my registry. You could use a registry cleaner program but I'm loathe to advice that because they can seriously mess up your system.