1) Yes, though there are some caveats to that. I believe your CNJ has to be the same level (or higher) as the quest in order to turn it in. May be a 5 level buffer. Like a Lv1 CNJ cashing in a Lv6 quest because they are within five levels of each other. Don't quote me on the levels. I just know you can switch, but I'm not 100% sure on what the lvlrequirement is.

If you just started, try to plow ARC up to 15 so you can get an airship pass to another city. Once there, you'll have access to a bunch of low level starter quests that are great when leveling a new job. That's what I'm doing with MRD. I purposefully am leaving all of the low level quests in Limsa alone for when I'm ready to level my MRD.

2) Yes. Each class has its own hunting log that is independent from the other.

3) Quests (aside job specific quests every fifth level) do not care what your current class is when you pick it up, or when you complete it. So long as you meet the level requirement. Again - whatever that is.