I’m also wondering would shake a woman’s hand? Personally when I’m in that situation I say “sorry, religious reasons” and they’re cool with that, my last professor who is also a dean of the college that I went to tried shaking my hand and I told her that and she was surprised because other Jewish men have.
Additionally not everything is black and white so you can’t say the Halacha follows the tzitz eliezer etc, because if I try hard enough I could find someone that assurs