1. Second Lieutenant is the maximum rank at this time. Future patches/expansions will add more.

2. Exp is simply wasted at level 50. It's not saved, not converted to anything. There are quests you'll want to do for their other rewards (reputation and tomestones from beastman dailies, unlocking content such as Good King Moggle Mog, Crystal Tower, relic weapon, dungeons, etc.). If you have lower level jobs which meet the level requirement for any given quest, you might as well use those jobs to do the quests.

3. Just mobs you kill.

4. There isn't a single dungeon in the game that doesn't require a quest to unlock. Once you've done the quest(s), some of them also have item level requirements before you can actually queue for them.