Timur, Emin'in babası. - The subject is Timur, at this point. You're presenting him to the asker.
Emin'in babası Timur. - Just a casual way of saying "Emin's father is Timur."
There are NO difference. You can use both of them.
But "Kim Emin'in babası?" doesn't sound natural. It's not 100% wrong but like I said, it sounds weird.
Although inversion forms are used in Turkish, the question sentences shouldn't be upside down. Otherwise, they don't sound right.