Are character names unique on each realm?

Yes. In order to promote individuality, World of Warcraft features an advanced naming system allowing players to create a unique name for their characters on each Realm.

Can Horde and Alliance characters gain faction points that allow them to utilize each others' cities?

No, unfortunately at this time we do not plan for you to be able to do this.

Can I play this game with a joystick or gamepad? Currently there are no plans to offer direct support in WoW for the use of a joystick or gamepad. The controls in the game are based on a mouse and keyboard interface.
Can I name my mount?

No. Mounts cannot be given names at this time.