Query fields are not stored in the database. It simply takes the result of your database query supplied to the field and displays it on profile. You'll need to include that query in your export query as a sub query if you want its results included in your export.
Is there a way I can trigger the auto action to run once for all users?
Yes, that's what the Internal Users trigger does. You'd then use the Internal Users URL provided in CB Auto Actions > Parameters to run all actions with the Internal Users trigger.