I just got a new PC and freshly downloaded DAO and DAO-Awakening from Steam. No all of a sudden Awakening is not showing up in the new campaign selection. Do I have to repeat the main game or is there something worng with my installation? Awakeing is not showing up anywhere though Steam insists it is installed.

Have you tried simply launching Awakening from Steam? Also, make sure that you are logged in to EA through the in-game client. Check your downloadable content section to see if Awakening has been installed or not.   
 
Other than that you could try to verify your steam cache. 

Or maybe try launching Awakening from the steam folder.

Well yeah. I actually always start games through Steam. I starts up and (although I dismissed that) always shows the codes. I even added th codes on the Bioware-Website (linked to my EA-Account) and bought the Darkspawn-DLC just to test it. It shows absolutely nothing in the DLC menu.
 
How do I verify?

Hum, the data seems to be alright.
 
All addons are activated on the BioWare-Website under the same e-mail-address I used to log in in-game.
 
I installed the firewall exception using the configuration utility.
 
Still no change.

Christ!

Ok did this stuff:
 
http://forums.steampowered.com/forums/showthread.php?t=1196144
 
I got some error messages - that the program can not access my user/Documents, but Awakening is available now. Not the DLC stuff though.

Awakening is launched through the "other campaigns" option in the game's main menu. 
 
You've been launching the game fine, but looking in the wrong place. 
 
Edit: Correction, it can also be launched from the "New Game" option. I've only done it through the other campaigns menu, so my bad.

Have you downloaded the other dlc from the bioware social site or are you thinking the game did it for you?  I'd recommend manually downloading and installing from the site if you haven't.