Is this Hollywood or did the firing of the guns carry through to the
controls and cause stick shake as the movies depict?

Hollywood thing. I've asked about this from various Finnish war pilots
and they haven't anything like that. For example Colonel Pokela on the
contrary described, that the Messerschmitt 109 fighter was "steady as a
train" when shooting.

jok

Take a look at the instruments on the panel when they do that. No matter what
the airplane is doing on the screen the indicators always show zero RPM,
straight and level, field elevation, zero climb/dive....etc.

Dan, U. S. Air Force, retired