This is a fantastically detailed response--very commendable! I wish there were a lot more detailed investigations of this sort.
However, I think a large majority of your points range from uncharitable to just plain wrong. I detail below. Despite that, I suspect that your overall conclusion about Dawkins may be correct, but the high error rate makes me more doubtful rather than more certain.
If Dawkins were ever sensitive, I'd say that he needed to be more sensitive, but that's just not his thing. He very likely needs to understand the issues better so he could use more accurate framing, though, as I do think there's a good argument that the modern understanding of his statements is problematic enough so he should pick different words. But anyway, on to (some) details:
(A) Regarding Dawkins' tweet about courtesy, I find your argumentation wholly unconvincing. If you tell me, "Call me Tuck," then out of courtesy I will call you "Tuck". It's very simple: you say you want such-and-so; it is easy for me to do ask you ask; and so I do it. It does not imply that there are no other reasons to do it (e.g. you don't want to be mistaken for Tucker Carlson, which could be very dangerous in the wrong company). It does not imply that it's not the normal, proper thing to do: saying "thank you" is also a courtesy but nonetheless it is rude to do otherwise. You correctly locate "courtesy" between "humor" and "respect", but overstate the inference. This is not a two-alternative forced choice reality. Dawkins can neither humor nor respect (in a deep sense): he can simply be courteous in this regard and leave it at that without deeper judgment.
(B) Regarding English being Dawkins' native language, you make a wholly unjustified inference that he's saying he's infallible. Nonsense! He's simply reaffirming that he meant what he said, in the conventional use of the terms. And he certainly didn't claim that his intent is what matters...in fact, that's the exact opposite of what the natural interpretation is. His claim is: there is such a thing as conventional usage of terms; I am well aware of such usage; and I have followed it properly. Frankly, I'm baffled that you even thought to use my-interpretation-only as an argument here. It might be a fair criticism regarding his claims about his intent, but not here!
(C) Regarding signing an affirmation that "trans women are women", your justification is complete rubbish. As stated, the requirement is to affirm a belief, not to agree to a way of acting. Any halfway competent person in HR ought to be able to tell the difference and phrase it appropriately. ("Treat other employees with respect", not "Have respect for other employees.") It is in exactly the same vein as loyalty statements to state or country, affirmations that you reject Communism (c.f. McCarthyism), and various other thought-policing of that nature. This has, or ought to have, consequences. For instance, technically it should be impossible to both accept the core Critical Race Theory approach (e.g. question the liberal order) and be funded by the state of California (including being a professor at its public universities; California is constitutionally obligated to uphold the liberal order). In practice, everyone just signs the loyalty oath these days, but it wasn't that long ago that disloyalty was a big issue.
It is possible that Dawkins was taking the directive out of context and it was actually an affirmation that one would follow the company policy to speak of and treat trans women as women. But without knowing what he had in mind specifically, it's also quite possible that it's not this. So your whole complaint boils down to assuming that Dawkins is motivated by anti-trans sentiment rather than anti-thought-control sentiment. Though you could be right, you shouldn't expect your stated target audience to believe you, given that Dawkins has a long history of opposing thought control.
(D) Almost all of your ten points about the Dolezal tweet are at best extremely uncharitable.
1. To contrast anatomy/genetics with societal categories and stay within a tweet's length, you can't use both man/woman and male/female. Your self-perceived gender (or sex) may not be a choice but how you identify is--historically, lots of trans people have been forced to identify differently from how they feel they really are. Dawkins is clearly setting up the two as a parallel, as observed from the outside, since we can't get inside the heads of the people making claims. So I think there is no particularly strong assumption here on Dawkins' part about the "right" answer. The charge that Dawkins is not being sensitive in the way that people who champion LGBTQ+ rights tend to be is true, but despite his formal courtesy, Dawkins is frequently insensitive to basically everyone, so that's not much of a guide to anything.
2. You mischaracterize him as saying that the internal feeling of gender identity is a choice. He's talking about how people present themselves (necessarily, given the parallel he's trying to draw). It is fantastically insensitive.
3-5. I think Dawkins is literally using the word "literally" the literal way. If X literally is Y, it means that not only X is said to be a Y, but it is a Y in every relevant way (not in-name-only, not kind-of-Y-ish, not Y but also this-not-Y-thing). There's no ambiguity here. He just means the idea that identifying as a woman is sufficient to be a woman in every relevant aspect; and identifying as a man is sufficient to be a man in every relevant aspect. Note the language: "they literally are what they identify as" (emphasis mine).
8. You wrote thousands of words villifying Dawkins for even asking leading questions of this sort, but say that villifying is a bit strong for outright denial? That's...a little hard to swallow? Also, job loss is a pretty severe consequence...and it's not like Dolezal is the typical case of someone claiming a different race than you might think they ought to. So I just don't get this objection at all. You'd need much better support to show that the reaction wasn't comparably strong given comparable media coverage, or that you wouldn't have comparable (progressive) media coverage given a comparably scandalous case (e.g. someone teaching trans history and involved in trans rights who was born female but was posing as a trans woman; or who actually themselves denies that identifying as female is enough to be female).
Note--the text is too long for me to get to everything, but the other sections have more flaws of the sort I've already discussed, along with a scattering of good points, including, for instance, that Dawkins can be expected to be cognizant of how his words will be used/misused. I'll skip to the end.
(E) You characterize the situation as "trans people are trying to survive" which suggests that Dawkins is perhaps making these comments to cause trans people to be killed. That is spectacularly ungenerous, given Dawkins' very long history of stridently opposing people being killed on the basis of beliefs (religious conflicts).
Furthermore, your numbers simply don't suggest it ought to be an overwhelming concern. I'm just going to trust you for now that your numbers are correct, because the flaw is with how you use the numbers that you report.
With 0.6% of the population now identifying as trans in the U.S. (https://news.gallup.com/poll/329708/lgbt-identification-rises-latest-estimate.aspx), 30ish murders puts the rate at around 15 per million, which obviously is 15 per million too many, but is about 25x lower than the population-wide murder rate in Chicago (and ~100x lower than the murder rate for black men in Chicago). So either you're undercounting by a lot, or you think Dawkins has a huge audience in Portuguese and Spanish speaking countries, or this is actually cause for appropriate concern but not different treatment than we would give to people who live in major cities. So: absolutely we should decry anti-trans violence and the fomenting of hatred against trans people. We should also do something about Chicago, Oakland, Detroit, Atlanta, etc., and the same logic you're using against Dawkins can be used tenfold against the "defund the police" slogan.
The bottom line is that it's a very poorly structured argument, mostly because the data you present doesn't jive with your characterization very well, especially in the dismal reality that many of us live in.
How I wish we would be so blessed that murder rates of 15 per million ought to be near the top of our list of concerns!
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People's rights are important, especially when under threat. Trans rights are no exception, and they are under threat.
But this only makes it more important to have good arguments, and also to build relationships with natural allies rather than antagonizing them.
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My read on Dawkins is that he's primarily concerned when ideology trumps reason. Most people who are ideologically driven believe that their ideology is for some greater good: you should tolerate free speech, but not this speech because it is for the greater good. Mob justice is bad, but not this mob justice because it is for the greater good. Burning people alive is bad, but not for this infidel at this stake because it is for the greater good.
Unfortunately, a lot of social justice movements sound very much like ideology a lot of the time: the arguments used most often are mostly emotional, not rational; there are certain core beliefs you're not allowed question (without excommunication, anyway); and opponents, or apparent opponents, are considered evil and are treated with neither compassion nor respect. People act this way when they're passionate and part of a group, regardless of whether they're right or wrong on the substance. None of these things are likely to ingratiate the movement to someone like Dawkins even if he agrees with the goal to an extent; the methods do not excuse the goal. And the AHA response just highlights the ideology. You wouldn't expect Sam Harris to have much patience with this either and...guess what!...he doesn't. Ayaan Hirsi Ali? Nope. Dennett? Nope. Goldstein? Nope. (Evidence for most of these at once, plus a fully justifiable criticism of Dawkins: https://www.youtube.com/watch?v=sAy2GOv9fd8)
So I think you have a really substantial task if you want to disentangle Dawkins' views on trans people from his views on trans advocacy. I don't think you've come anywhere close to doing that even in a long piece such as this.
Furthermore, even if you don't admit any distinction between the two, and just are assessing "Dawkins: friend or foe?", you would do much better focusing on a few strong arguments than diluting them with weak to specious arguments as I believe you've done here (see details above).
Anyway, I'm undecided as to whether unleashing all of people's passions, ideology and all, is the better way to achieve social justice than running on a cool evidence- and reason-based approach. Throughout history, we mostly have seen the former...and usually have mucked up at least some part of it in a way that we wouldn't have if we'd had clearer heads. But does the latter way even work? The examples are few to nonexistent. (Banning CFCs, for instance, was of this type, but it's a stretch to call it a "social justice movement".)
Nonetheless, if you take the former approach and find New Atheist types making simple strident arguments against things you believe you adequately addressed...well...don't act surprised. If you want it to go away, I suggest (i) get your arguments and evidence in favor of your position really well explained and really airtight, if you can, and (ii) stop throwing petrol on the fire by using more breath on attacking their character than on their correctness--just keep hammering home the arguments.
(Note 1: if you haven't already committed yourself to viewpoint diversity, your arguments are probably inadequate because there will be too many assumptions you haven't challenged. Even if you're right, expect to have to go back and forth with a debate-minded person of opposing viewpoint to iron things out until the arguments are not full of holes.)
(Note 2: yes, I know you linked to arguments for some core positions. You can reuse the same arguments if they're strong enough. But point (ii) still applies.)