Unfortunately, at the moment you can't really see which quests you've completed. You have to know how many stages each quest has and then check each quest one-by-one in the quest selection menu (it shows the quest stages as a drop down menu). The devs know how bothersome this is and plan to make it easier for us to know which quests we're for the reset.
As for the second part of your question, what's a gab?