Yes, it is possible to have both a class-specific mount and a mount purchased from a vendor. However, at this time, only warlocks and paladins have class-specific mounts available to them.

When a hunter tames a new pet, its default name will be the type of beast it is, such as "Boar". You can set a new name by right-clicking the pet’s portrait in the upper-left part of the screen. You can only rename each pet once, so choose carefully, and remember that the naming policy also applies to pet names.

Each individual warlock will always summon the same imp, the same voidwalker, and so on. A name is randomly assigned the first time that a particular demon is summoned, and that name is permanent and cannot be changed.

"Vanity pets" such as cats, snakes, cockroaches, and Collector’s Edition pets cannot be assigned names.

The ability to shape shift is first learned from a quest you get at level 10. You will learn more shape shift forms at later levels

Yes, you can be in combat and attempt to tame a beast. In fact, when you start to tame, the beast will automatically attack you while you channel.