Pretty much you can alternate classes as you like. Some require a particular category of class (like needing to be accepted and turned in by a combat class rather than crafting class or vice versa) but you can alternate within that category, and your class quests / job quests have to be done by that specific class or job. The turn-in XP goes to the class you are when you turn it in, regardless of which class you were when you completed any of the quest's other objectives.