Learn Polish: Piotrze, czy masz dzisiaj czas? - Peter, do you have time today?

It is in the Vocative case (Wołacz), which Polish uses whenever you are directly addressing or calling out to someone. For masculine names ending in a hard consonant, the Vocative usually adds an -e to the end.

Where did the -rz- come from in Piotrze? The dictionary form is Piotr.

When adding the -e ending for the Vocative case, the final consonant of the name often undergoes a spelling change to soften it. Here, the hard r in Piotr shifts to rz before the -e. This is the exact same spelling change you might remember from the Locative case!

The verb masz (you have) requires the Accusative case. Why didn't the word czas change its ending?

Czas is a masculine inanimate noun, meaning it is a thing, not a living person or animal. For masculine inanimate nouns, the Accusative case looks exactly identical to the basic dictionary form, so no special ending is added.

Do I always have to use czy to ask a yes/no question like this?

Not at all! In spoken Polish, you can completely drop czy and just say Piotrze, masz dzisiaj czas? As long as you raise your intonation at the end of the sentence, people will know you are asking a question.