A) Yes, you need to buy the game. That purchase will give you 30 days of access, and during that 30 days (or any time between purchase and paying an additional subscription fee for the first time) you can put in one of those recruit codes. You only get the bonus items from that recruit code once you pay an additional subscription fee. (If paying by credit card direct to SE, you can only pay for time once you need to, so it won't be until after your initial 30 days are complete. If you pay by game time card, you can do that any time after purchase of the game.)

B) Once you get to level 15 (where dungeons start) you could level up a particular class/job just by dungeons and one class quest each five levels. That's only if you're using another class to do the rest of the main story though. Most content in the game is unlocked either directly through the main storyline or at least requires a certain amount of the main story before you can unlock it. The main story itself alternates, with the majority of its content occurring in open-world quests, and some of it occurring in dungeon runs, so you can neither play completely solo nor completely in dungeon groups. You could, however, do the dungeon content as conjurer/white mage and the solo content as thaumaturge/black mage or some such combination if you want to.