Flying ????

[Risvertasashi]

Yes, you do have to finish a zone before you can fly in it.

[Noitems]

Unlocking flying for the older zones is great as many of the job quests between DoW/M and L/H have to go visit those areas multiple times. Especially when you get to the gathering jobs you will find yourself constantly visiting those zones for the quests but also to work on leves for the materials that are there.

Also if you want to do hunts it's heavily regarded you unlock those currents as hunt trains will go visit HW zones for A/S ranks. If you're on foot/mounted on ground you may not get there in time for the pulls as opposed to flying.

[Pepsi_Plunge]

Hunting, Gathering, Tribe Quests, Jobquests, Anima Weapons, special Fates and other sidequests. Just because you are finished with the Main Story in said zone doesn't mean you will never have to go there again.

[EveK]

There's an "Aether Currents" option on the Travel menu (off the main menu) that specifically details how many quests and field currents you have/need in each area. You don't need to do all quests in the area - just the few that unlock currents.