In the ability database, some of them are learned from leveling and some of them you buy with marks. I believe so, anyway. I bought Heavy Swing on my Marauder long before I hit rank 20 so I don't know if I would have wound up learning it automatically anyway.
Couldn't answer your question about class quests though.
Naiya | Mithra | Windurst | Rank 10 | 1st Lieutenant | Ragnarok
WAR: 75 MNK: 75 SAM: 66
RotZ: O | CoP: O | ToAU: O | Apoch Nigh: O
Naiyah | B.Elf | 85 Rogue | Stormscale
Doubleday | B.Elf | 85 Paladin | Stormscale