[Michael Cule]

Yes, qualifying as a full member of the Mages Guild of Tredroy requires you to enter the building without asking the Doorkeeper to open up for you. You don't actually have to overcome the Magelock (at least in my Tredroy): getting in via teleport (to the small area behind the door that isn't protected against that) or walking through the walls in insubstantial form is allowable. (Again there are protections against that sort of thing in most areas.)

That test applies when you're coming in as a trained journeyman from another guild or when you're an Apprentice from the Mages Guild of Tredroy proving you qualify.

To become an apprentice at the Guild is another matter. I think it primarily involves finding a Master willing to take you on and train you. There may be exams required by the Guild but you still have to convince a Master.

What would they test?

Your level of Magery. If you're only a Mage-0 they are going to point you towards working with a hedge witch or other discipline.

Your general level of education. Papers to test literacy, numeracy and general knowledge.

Your intelligence... But I'm not sure how they would do this. Puzzles, logic problems, practical tests of the sort that officer selection boards like?

Your (or your family's) financial resources. There may be scholarships available. Or not. Bribery might work. Or not.

The one occasion when this became a concern in my games the applicant was a uniquely gifted Northman and the contest was between the Masters for who would take him on. But those are the sorts of things they were looking for in an apprentice IIRC.