Personal pets? No.
Beastmaster class? Yes, with temporary pets, just like FFXI.
A monster battling arena, similar to the one in FFXI(does anyone remember that?) would be a fun time killer. Who doesn't love Pokemon?
As for personal mounts, I could care less. A chocobo would be useful every now and then, but I don't want to twiddle my hands and have my Chocobo magically poof under me. Kweh McWark can chill in the Uldah stables until I'm ready to ride him, as far as I care.