Yes. You can complete your first class change quest at 19. Being level 20 is meaningless for this quest. You *must* complete your class change quest first.
Yes, any character level 19 or above in an academy can do the quest.
There is one caveat, however, and that is you must have a sponsor to start and finish the quest.
Contact one of the players in your main clan to have them sponsor you. They will need to be there for parts of the quest. In addition it is the sponsor who has to giev the crystals to the blacksmith before he will make your armor.