Learn Croatian: Što je danas ručak?

No, you cannot. The word je is a clitic, which is a small, unstressed word. In Croatian, clitics follow a strict rule: they must always go in the second position in a sentence. Since Što is the first word, the clitic je has to lock into that second slot right behind it.

Could I change the word order and say Što je ručak danas?

Yes, absolutely! Once you have the first word and the clitic locked into their required positions (Što je), the order of the remaining words is very flexible. Both Što je danas ručak? and Što je ručak danas? are perfectly natural and mean the exact same thing.

What case is the word ručak in here?

It is in the nominative case, which is the basic dictionary form of the word. When you use a form of the verb to be (like je), it acts like an equals sign between two parts of the sentence. Because of this, it doesn't force the noun into the accusative case like an action verb would, so ručak stays in the nominative.

I sometimes hear native speakers say Šta instead of Što. Are they the same?

Yes, they mean the exact same thing. Što is the standard Croatian word that you will learn in courses and see in writing. However, Šta is a very common colloquial variation that you will hear constantly in everyday, casual speech.