Are the non-regular languages closed under reverse, union, concatenation, etc?
Non regular languages are closed under reverse, because $L = (L^R)^R$. Same is true for complement.

Non regular languages are not closed under most other basic operations though. Consider, for example, that $L \cup \overline{L} = \Sigma^\star$. Similarly, if $L = \{1^{x^2} | x &gt; 1\}$ then $\overline{L} \circ \overline{L} = 1^\star$.

There is no complete list of operations on languages, so a complete answer cannot be given.