Closure property and undecidability
I don't think closure property implies decidability. Consider r.e. languages and intersection. R.e. languages are closed under intersection, e.g., if $L_1$ and $L_2$ are r.e. languages then $L_1 \cap L_2$ is also r.e. but may be undecidable.

On the other hands, CFLs are not closed under intersection. However, a language which is intersection of two CFLs is not CF but may be decidable. For example $L_1=\{a^ib^ia^j |i,j \geq 1 \}$, $L_2=\{a^ib^ja^j |i,j \geq 1 \}$ are both CF, however $L_1 \cap L_2 = \{ a^ib^ic^i | i,j \geq 1\}$ is not CF, but it is still decidable.