Could you expand on your comment about a

"general pay gap" between men and women? I have seen far too many studies to take this at face value anymore (even the WaPo has debunked it on numerous occasions - I could pull up the link but my internet connection is terrible where I am right now). The 20-30% "difference" in pay was a stat that was pulled ~4 decades ago, and simply took all the earnings of men across jobs and compared it to all the earnings of women across jobs, and then compared the total difference. One doesn't need to take a statistics course to understand the flaw in that methodology. In the various studies I've read, the current differences (which are much smaller than the 20-30% of decades past) are mostly made up of personal choices. It is apparent that men choose more careers in STEM fields which tend to pay more, men choose jobs that typically require longer hours, men are more willing to relocate for their jobs, and men tend not to have as much time out of the workforce due to child-bearing and child-rearing (fair or unfair it's true). We can talk about society and expectations for men and women and how that impacts career choices, but that's an entirely different discussion. I also read a study conducted in multiple Scandaiavian countries where things are at least perceived to be far more egalitarian than here, and the choices men and women made were even more pronounced than they are here. Women tended to go into more "humanitarian" fields (nursing, social work, etc.) where there is far more human interaction but lower pay, and more men went into the higher paying STEM fields. So while this "gender pay gap" is often repeated by certain politicians looking to gain favor, that doesn't make it true. So I would need some evidence of equally qualified and experienced men and women in the same roles, doing the same work, and with the men making more (and not in sports where it's revenue-based please). We have laws against that type of behavior as well.
I have seen it both ways, and it can be subjective. Additionally, and I'll admit it's anecdotal, but at my large employer and my large employer clients women are specifically sought out for some of these more financial-related roles to get more representation, and often given an inherent advantage in that process.

Regarding your paragraph about LGBT people, I would make a counter argument that men participating in sports that require physical attributes is unfair to women who are not born with the inherent biological advantages that men have.