As far as I remember, I did have to enter a CD-Key when I installed the game.

If your account is accessible with your mail and your password again, you don't need the cd key.
I guess they won't tell you your cd key as it was only required for binding the sc2 license to your account. You can install and play it without knowing your key. At most you would need to tell them your key to unlock your account.
Especially I wouldn't ask for your own key when you want to recover your account....

You entered a key to register SC2 on your Battle.net account. You don't need a key to install the game, though.

I believe the last time I tried to download StarCraft 1 from my Battle.net account, I did have to get my CD-Key. I guess it's just not the same with SC2.

They changed a lot since SC1. Everything is about adding a game to your account since it all has to be played online.

Problem solved.
I had forgotten to try a certain combination when entering my name...

If it's still relevant, when you register an authenticator, you register the serial number of it, and Bnet then adds a field on the login screen for the authenticator key. When you press the cute little button, it uses the time and date as a seed and performs some flajoobity mathematical equation using the serial number, generating a key. After a while, the resulting key expires, but a simple press of the button repeats the process. Assuming the process is still the same, of course.