Learn Finnish: Mikä tämä kuva on?

Yes, both mean "what", but they do different jobs. You use mikä to ask about the basic identity of a whole, countable thing (like a picture, kuva). You will learn to use mitä later for uncountable things (like coffee or water) or when asking about actions.

Why is the verb on at the very end of the sentence?

In English, we swap the verb and subject to ask a question ("What is this picture?"). In Finnish, starting with a question word like mikä already makes it a question, so the rest of the words keep the normal subject-verb order. The subject (tämä kuva) comes before the verb (on), making it literally "What this picture is?".

How would I change this to ask "Who is this?"

Because you would be asking about a person instead of a thing, you would swap mikä for kuka (who). You would also drop the word picture. The sentence structure stays exactly the same: Kuka tämä on? (literally "Who this is?").

Do tämä and kuva have to stay next to each other?

Yes. Tämä (this) is a descriptive pointing word, and just like in English, it belongs right in front of the noun it describes. Together, tämä kuva (this picture) acts as one single, unbreakable block in the sentence.