1. Mounts let you move slightly faster than sprint without any cooldown.

2. You do not select your mount. You may however acquire different mounts by completing quests or through events.

3. No, and I do not believe there are plans for an in-game store. Closest thing to this is probably the Coeurl mount and Chocobo Behemoth barding, both of which are items exclusive with the Collectors Edition of the game. If you want to purchase it, buy the digital CE upgrade or CE hard copy.

4. it is very unlikely you will see flying mounts. Although who can really say? Maybe you will in 2 or 3 expansions down the road. I just wouldn't say it'll be anytime soon.