The biggest difference between us is that I believe all humans should be bound by the same rules and reason, you seem to think there should be different rules and reasons based on gender.

You are wrong about that. I too want that all humans are be bound by the same rules and reason.

However, I believe that in order to archive true equality women need to take on more responsibilities and accountabilities and men should get more rights.

1. Men need to register for selective service to vote. So should women.
2. Women can use safe heaven abandonment to rid themselves from the responsibilities of motherhood. Men should be able to to the same.
3. Girls genitals are protected from mutilation. So should be boys genitals (and intersex for that matter).
4. The law describing rape should be written as clearly gender neutral and clearly acknowledge made to penetrate as a crime.

I only want for men what women already have.

About 1.: Google „The Order of the While Feather“ — You can't abolish the draft. If you don't have a draft in law then something worse will appear in it's absences through the means of peer pressure.

About 2. & 4.: For those you could go the other way as well. No safe heaven abandonment and there is no real reason to have a separate rape and sexual assault law. One law could cover both.

I'm looking forward to the cognitive dissonance needed to justify inequalities to the disadvantage of men