Yes, it is possible. A moderator may dismiss a flag as valid (you see an increase to your flag weight) or invalid (you see a decrease to your flag weight). I'm not entirely sure if there's any way to dismiss a flag with no effect to your weight, but I don't think so.
(I share your luck with p.se flags. I myself have been unable to submit a valid flag to that site. I've taken that as a sign that I don't understand what is and is not on topic there, and have thereby stopped flagging on that site)
To your edit: yes, they can. Taking action iscan be separate from marking mod flags.