How the closure properties of the formal languages dictate decidability of their problem
If a family $\mathcal F$ of languages is closed under some operation, say intersection, then you are right that it is decidable whether the intersection of any two particular languages from $\mathcal F$ also belongs to $\mathcal F$. The answer is always Yes.

The converse, however, isn't true. Consider the family $\mathcal F_m$ of all regular languages which are accepted by some DFA having at most $m$ states, for some parameter $m$. When $m &gt; 1$, this language family isn't closed under intersection, but given any two languages from $\mathcal F_m$ (in any reasonable encoding), it is decidable (depending on the encoding, efficiently) whether their intersection belongs to $\mathcal F_m$.