My reply above was aimed at addressing part of your question #2 - the desire for SOME trans people to live out traditional or stereotypical representations of masculinity and femininity in light of the fact that they are socially and culturally constructed. We as humans conform to gender norms for survival, acceptance and belonging - trans and cisgender people have this in common. The greater any of us veer away from these norms, the more we run the risk of reduced access to social privileges and safety.
In order to accomplish this "fitting in", some (not all) transgender people utilize medical interventions (your question #1) to more closely resemble the biological sex that society (and even we as individuals) expects someone to look like as a given gender. We use medical interventions for a host of things that are not strictly essential for survival - in this case many might argue that fostering one's ability to most comfortably live out their gender identity, to allow them to more fully function and thrive in society, rather than continuing to live in their own personal hell and increasing their risk of suicide, is about survival.
I love science; the various fields of study are wonderful things. I continued to be amazed at their advancements and what we as a species are accomplishing, but these fields are evolving - they don't have all the answers. We can't biologically nail down what the human psyche or consciousness is precisely, or how it fully operates. We don't have a perfect biological or sociological or psychological explanation as to why some people have purely opposite sex, purely same-sex or mixed-sex attractions. Does that mean these things don't exist? Do you want to try telling the world that homosexuals don't exist? Does that we can't let a person count themselves as heterosexual or homosexual or bisexual? That we can't gather data and statistics about the predominance of these identities?
Gender and biological sex are not one in the same, and yet they are closely linked. We as cultures and societies have organized gender roles and expectations around genitalia and body structures (which is part of why some trans people look to alter these to match the gender they identify with). But the expectations and roles are not biologically prescribed in and of themselves - concepts of masculinity and femininity vary across different times, geographies and cultures. Historically, and for simplicity sake, we have categorized biological sex and gender as the same thing. And whether we choose to tease them apart will ultimately depend on the purpose of analysis. If we're trying to look at the demand for social or recreational programs for instance, then data could be collected along gender lines. If we're looking at particular diseases or physiological processes, then it might make sense to look at the DNA level (ie. chromosomal biological sex)........... or it might make sense to look at the hormonal levels (which might lump transmen with cis-men and transwomen with cis-women). If we're looking at HIV transmission risk rates and reduction strategies, then we need to look at more than just the traditional male / female divide - we need the additional transman and transwoman categories.
Right now the biological community can't even concretely define what we mean by biologically male and biologically female. I can't provide the link directly (whenever I try the comment doesn't post), but look up "Sex Redefined" by Claire Ainsworth on nature.com. It speaks to this directly, and points out that the biological and medical fields might need to possibly consider more biological sex categories, to treat biological sex more like a spectrum or to slice and categorize it across multiple planes.
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