Emily
The OP wishes to be informed. Is that a bad thing. How often do we see the equivelent of "you should have known" printed here. Well, you cant have it both ways.
One of the first questions to the OP was what passport. Very relevant question and a second might be what ethnicity ... unfortunately.
My Asian expat friend here in the EU is waiting on the delivery of her renewed residency card and in the mean time has had to cancel a trip to Vienna. i asked why, its Schengen, no one will check. "I am not you" was the answer. If an opportuntiy exists, I will be checked.
Same with you flashing your residency card at the border. Your experience will be different, so its hard to compare with even a US passport alone. By the way, I got spanked in Frankfurt for not presenting the residency card at immigration. I dont blame him. I knew better. Just lazy.
Same with, "should I carry my passport when I am out and about in Vienna?" Depends on what you look like, what language you speak and how you dress.