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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.