English doesn't have "case". There are remnants of the cases used in Anglo-Saxon, but there is no comprehensive case system in modern English.
So "mine" is a possessive pronoun.
English doesn't have cases, so there is no genitive case of "that". And while you can form the possessive "Everyone's" with the clitic 's there is no genitive form. You can use "of" to form possessive expressions with "that", but this is very rare in practice (and so learners should normally avoid it)
The Old English neuter genitive of Þæt was Þæs but that is absolutely irrelevant to modern English