As for part 2, of your question, that is a tougher nut to crack. It's not just believing in gender equality. It's recognizing the deep, deep roots of gender inequality, which has tentacles in so many things: notions of biology and essentialism, cultural norms, etc.
I suppose my opening wish would be that people who do consider themselves progressive or believe in gender equality, or any equality, would be willing to take the hard look sometimes required to understand just how engrained some of these notions are. How do we perceive things differently coming from a woman versus a man? From a person of color versus someone white? It's an uncomfortable conversation because no one wants to wind up looking like a jerk. But we can't move forward without having it.