Concerning Rav Moshe’s shita on shaking hands with the opposite gender, there are obviously exceptions; read the following excerpt from a personal direct experience with Rav Moshe Feinstein ZT”L:
[unobservant] like me has a yeshiva bochur [student]
Rav Moshe finally looked at me. I was mortified. My aunt was addressing him irreverently. She was joking with him. She had called me Ikey, not Yitzchok, or even Isaac.
[religious]