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][1]:
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.


  [1]: https://dx.doi.org/10.1090/S0002-9939-1995-1260168-2