You can get vanity non-combat pets but none that are useful while fighting.
Rogues can have noncombat pets and guardians. Guardians can assist in combat, but are not controlled directly by the summoner. These are different from the pets available to the Warlock, Hunter, and Death Knight.
you need to be a hunter, to name your pet, if you are you can name him by pressing "c", going to "pet", then beside name, type the name.
combat pets like warlock minions and hunter pets automatically learn most of the spells they use as they level. other spells can be learned through the pet talent points.
There is a such thing as ocean hunting in WOW its just not called ocean hunting. its called swimming underwater beating up everything you find. There is a hunter class, but they have no special abilities regarding water, nor special underwater pets.
Hunters ---------- locks and deathknights
First off, the Hunter class can tame a large variety of creatures as pets. Hunters have an ability called "Beast Lore", with which they can check any creature and see if it can be tamed. Hunters can also tame a large number of rare creatures, which spawn rarely around the world - usually, this involves being a Beastmaster speciality Hunter, because these are often exotic creatures. The full list of tamable pets is too large to reproduce here, I would recommend checking fansites like Petopia or a Rare Hunter blog for a good list.
No, Hunter pets remain as they look when you tame them. For all other classes with pets the same goes, they look the same for all people with that class. In Cataclysm, Death Knights will get an ability that changes the way their Ghoul looks (on a cooldown) into something more like an Abomination, which will also give them new abilities. This is the only "customization" announced so far.
wow...
the T10 class gear
http://www.wowhead.com/?spells=-6 has a list of all the Non-Combat pets in WoW
Bow chicka wow wow ;) You know what I mean!
If its a mount, you dont need to feed it. If it is a hunter pet, it will eat fruit.