Kingdom of Knuffel

a loving but deranged white rabbit

I'm pretty sure a leopard is your familiar!
That, or a lynx.
Might be a lynx.

You have a baby goat

You've got an egret.
Its beak is made of mithril.
It might be stronger than the King.
Shhh, don't tell.

You have a hawk

No one can see your familiar, because it is invisible. Its species used to be perfectly visible, but it was so pretty it ended up having a constant visual hypnotic effect on everyone exposed to it, and the Alchemist at the castle eventually came up with a method to change that.... except the side effect was a little more extreme than expected.
To keep it from going extinct entirely (hard to do an animal's normal mating routines when suddenly the whole species is invisible), there was an intense kingdom-wide adoption drive to turn them all into pets with fully funded breeding programs so they could survive.
You ended up bonding to one, and here you are.

Your familiar is a cute koala bear.

An extra fluffy white cat

your familiar is a lynx.

I think you might have some sort of lovebird?