For question (1), there are many exceptions. For example, being superstrong is $\Sigma_2$ expressible, since it is witnessed inside a sufficiently large $V_\theta$, but this is stronger than strong in consistency strength, and being strong is $\Pi_3$. Similarly, being supercompact up to an inaccessible is $\Sigma_2$, but stronger than supercompact in strength, while being supercompact is $\Pi_3$. One can make many similar examples of very strong $\Sigma_2$ properties, by asserting that a strong $\Pi_3$ property holds up to an inaccessible cardinal. This reduces the complexity of the assertion, but is stronger consistency-wise.

For question (1), there are many exceptions. For example, being superstrong is $\Sigma_2$ expressible, since it is witnessed inside a sufficiently large $V_\theta$, but this is stronger than strong in consistency strength, and being strong is $\Pi_3$. Similarly, being supercompact up to an inaccessible is $\Sigma_2$, but stronger than supercompact in strength, while being supercompact is $\Pi_3$. One can make many similar examples of very strong $\Sigma_2$ properties, by asserting that a strong $\Pi_3$ property holds up to an inaccessible cardinal. This reduces the complexity of the assertion, but is stronger consistency-wise.

For question (2), I would suggest Vopěnka's principle as a commonly considered large cardinal concept that, if consistent, is not first-order expressible as a single assertion in the language of set theory. This axiom is typically formulated in second-order theories such as GBC.