Yes you sure do need permission for all special characters played overtly on SS. But thats pretty fun in the MoM since we have registry offices for certain 'special' things and you'll get the opportunity to choose between RPing out the registration process or playing as if they have been registered in the past. Think of it this way, if its something that would be illegal, or something that requires registration in canon (animagi, werewolf etc) then someone wouldn't be hired if it was revealed that they weren't registered would they? (and trust me, our security peeps do thorough background checks it WOULD be discovered *evil cackle*)

Likewise with special pets, in canon many creatures require a license. The best thing to do is to PM the department of Magical Creatures Head, and find out if the pet you are intending needs registration, or if a plot could be worked out. (once you are an employee of the MoM) Some creatures are illegal to own, others are illegal to breed, and still others can be owned but certain charms must be regularly applied to be sure the statute isn't breached.

Also, don't forget the MoM is a place of work, and pets wouldn't be allowed to wander around willy nilly, and in DA too, dangerous pets would not be permitted to wander around, and some shops might not let your character bring a pet in at all

Hope this helps!