Some of my pets suddenly disappear without any message. Sometimes it can be found at a random player's stable even though I haven't gifted or sold it. While other times it disappears completely.

Thats crazy!!! I have NEVER heard of that happing before!!!

Is it possible that someone hacked into your account? If thats the case, you might want to ask if your account can be banned and start a new one afterwards.

Still, thats insane!

I wonder if changing password will help.

Oh, yes do that. Does anyone else use your PC? If that is the case, you might want to also change your password on your PC so nobody can use it. And when you leave Beastkeeper, make sure you log out properly. ( little button called Logout right next to your name at the top righthand side)