If i would buy this version, could i then copy/install awakenings to my existing version Dragon Age Origins? I allready have all other DLC's.

I suggest you to try opening a ticket, or something of the sort, with Steam's support about using the DLC from other store with their base game. But I would hardly believe that it is possible.

Edit: some peoples gave some thoughs about a similar situation with Witcher 3: https://www.gog.com/forum/the_witcher_3_wild_hunt/steam_dlc_when_bought_on_gog.

But I believe that is easier to just backup your save file from Steam and play it with the GOG's DA:O.