I just finished vanilla F03, Is my character done for? What happens when you install the DLCs? Any order I should play them in?

Without Broken Steel installed, once you activate the purifier, it's Game Over. After installing Broken Steel, you'll need to reload a save prior to entering the Rotunda in order to get past the vanilla "ending" and into Broken Steel.

I would recommend playing Mothership Zeta first out of all the DLCs simply because of the difficulty. At levels above 20, many of the enemies can be nearly impossible to kill.