Does the vocative case in Czech mean that it is an endearing form of the name? Russian has this. For example the name Saša. A good friend or family member might call him Saška. Or is it a required case in Czech when addressing someone? Also, does formal/informal have anything to do with it?
What you are referring to in your question is known as a diminutive. Endearing "small" versions of names, which are informal by nature; ie. only for use with friends and family, or at least someone you know relatively well.
All of these names are conjugated according to the rules of the vocative case when addressing someone.. for example; Jakub = Kuba (vocative: Jakube, Kubo), Zuzana = Zuzka (Zuzano, Zuzko), Vladimir = Vlad'a (Vladimire, Vlad'o), and so on.. Many names have multiple diminutives, sometimes even denoting degrees of endearment; Jakub -> Kuba -> Kubiček (Vocative: Kubičko!)
Baseline: Vocative, always used; diminutives, only with friends. :)
Vocative is a required case in Czech when addressing someone, regardless of formality level.
"Kateřino, pojď do kina"