Is there a way to export custom profile questions? Either as a event report or just in general?
Yes you can export Custom Data. If those fields are used for Contacts rather than eg Participants it is probably easiest to do this via Adv Search. Find the contacts you want. Then - go to Actions and - select 'Export Contacts' and then click on - Select Fields for Export - then choose the fields you want to export - once you have done this you can Save your Export Mapping and choose it another time
If the data was stored on the Participant record then you might find this easier to do via Find Participants
If you are not sure if the Custom Fields are used for Contacts or for Participants (look in the Custom Field Set page at /civicrm/admin/custom/group?reset=1 and see what it says it is Used For.