1. Yes, but your CNJ has to be the some level as the quest or higher. Note that most quests require you to be a combat class (DoW/DoM), so you cannot turn them in as a crafter (DoH) or gatherer (DoL) class. Also, if you're doing a F.A.T.E. and switch classes, I believe your contribution resets and you have to start over, so don't try to level another class that way.

2. Yes.

3. Changing classes only changes your skills and class/job-specific hotbars (#1-3). Nothing else changs. Your quest log stays the way it is, though some quests have turn-in requirements like a level minimum (and class quests require you to be the class for the quest). If the quest can't be done as your current class, you should get a message saying so when you try to pick it up or turn it in.

Also, if it wasn't obvious, you can create gear-sets for your different classes on the character screen, and drag those gear-sets to hotbar keys. Putting a bunch of gear-sets for different classes on a hotbar that doesn't change when you change classes (#4-10) is a great way to quickly switch between them.