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As an answer to the bonus question: no, see K. P. Hart and J. Vermeer. Fixed-point sets of autohomeomorphisms of compact F-spaces, Proceedings of the American Mathematical Society, 123 (1995), 311–314: every fixed-point set of an autohomeomorphism of $\omega^*$ is a $P$-set. The same paper also answers the other bonus question: every $P$-set is the fixed-point set of an involution. Apply this to a $P$-point.