Can any WOW class have a pet?

Answer 1

Yes all classes can have pets, but it depends on what you mean.

You can get a companion pet from various vendors and from the Blizzard store

Warlocks get minions.

Hunters get tamed beasts.

Death Knights get guardians (like their Rise Dead spell), or a permanent ghoul if they are specced into the unholy talent tree.

Mages get water elementals if they are in the frost talent tree

All classes can also get guardians that lasts for a certain amount of time, which are usually received from completing quests.