Learn Finnish: Kuka tahansa kuningas kumartaa minulle.

While kuka tahansa on its own means 'anyone', you can also place a noun right after it to mean 'any [noun]'. Because a king is a person, we use kuka here to say 'any king'.

If I wanted to say 'any animal' or 'any book', would I still use kuka tahansa?

No, you would use mikä tahansa for things and animals. Kuka tahansa is strictly used for people. So 'any book' is mikä tahansa kirja, but 'any king' is kuka tahansa kuningas.

Does the word tahansa ever change its ending if the noun is in a different case?

Never! The word tahansa is a fixed particle that stays exactly the same. If the sentence required a different case, only the pronoun kuka and the noun kuningas would change, for example kenelle tahansa kuninkaalle (to any king).

Why is minulle in the allative case (the -lle ending)?

The verb kumartaa (to bow) requires the allative case to show who is being bowed to. You are bowing 'to' someone, so it takes the -lle ending.