So, by supporting a male-only military draft, you thereby support a double standard of autonomy favoring women to which men would not have access...

One never "supports" downsides. One tolerates them because the alternative is worse.

Why should everyone feel a "duty" to participate in sexual reproduction/procreation?

Did I say "everyone?" I said "healthy," but only as short-hand for having no special reason, e.g., homosexuality, not to reproduce. (I intend no connotation of "illness" on gays and apologize if I gave that impression.) Preservation of the species requires reproduction. Someone's got to do it, and unless you want a Handmaid's Tale world, a moral obligation of otherwise suitable free people seems to me the way to go. If you've got a better way, feel free to expound.

Aside from producing offspring and serving in the military -- are there any behavioral or personality characteristics that you believe should be exclusive to either men or women?

The entire notion of gender turns on preparation for the two fundamental roles I have mentioned. Cataloguing the differences adds nothing, as they can pretty much all be deduced from the basic roles. The Darwinian point is that complementarity is more adaptive than interchangeability. The rest is details.

Or are you suggesting that we discard "masculine" and "feminine" references entirely?

I want to keep those words. You're the one eviscerating them so as to make them obsolete.