I don't often post to public forums about topics like this, but from reading the FAQ I believe I'm supposed to point out poor arguments where they show up. Sorry I know this thread is very old but if I go around recommended articles on this site then I can't in good conscience not point out the number of logical leaps you've made in this response. Many of these were pointed out by An_Amazing_Login but she was understandably upset about the whole thing. I have no personal stake in the politics at hand, this is mostly just a rational rebuttal.
Firstly, I think its easy enough to point out this bad-faith argument-
"This is a very worrying example. For in fact there was such an operation and it may have even got a Nobel prize. It was called the frontal lobotomy and in an age before anti psychotics it was a very useful operations allowing schizophrenic and manic patients to live perfectly peaceable lives no longer confined to the asylums." While that's true, it's also functionally irrelevant to the topic at hand. Medical ethics have advanced rather far since the age of the lobotomy and clearly Mr. Alexander wasn't referring to a horrifically damaging procedure in his offhand example of why low-cost treatments to debilitating psychological conditions would be nice. I imagine the obvious counterexample would be to say "nobody thought lobotomies were unethical when they were performed on marginalized people and yet they still destroyed people's lives. Perhaps most doctors don't think sex-change operations are unethical now but in 50 years we will consider the cutting-off-of-penis surgery (not mentioning the converse because you seem fixated on this one) to be highly unethical as well. I would consider that a good-faith argument, and I think the easy answer of bringing up the virtual infinity of other medical procedures that could be considered highly unethical one day to be a good point to begin with in response.
Your argument on the proposed medical treatment for depression is somewhat similar. We already prescribe drugs for depression, so clearly medical ethics boards think risking (frankly horrific) side effects is in some cases worth it. I personally disagree, but then if I could actually, you know, see the side effects of the mythical hormones in question I would consider it. In reality, the side effects of hormone therapy on trans people seem to be passing my personal bar for ethical treatment standards, to a far greater extent than the prescription of benzodiazepenes or opiods.
Second bad-faith argument-examples of the rape of trans men and the question-would you want to send a man in possesion of a vagina to a male penitentiary? As the other commenter rightly pointed out, I don't think a man being raped is any better than a women being raped, so this argument doesn't really make much sense. Unless you're arguing that the presence of a biological female makes a man rape where he otherwise would not, which I would argue seems to be a problem with the man, not with the sex of the other person in the room. To put it another way, if your issue with transgenderism is that it will lead to people with vaginas being raped where they would otherwise not be, then I suppose giving all biological females penises and reproducing via surgical procedures would be a moral imperative that we should get on making affordable. Similar reasoning applies to the other way around, unless you think that women can't rape other women. In fact, your arguments on this subject boil down to "the inherent physical advantage that going through male puberty confers means that most men will have the advantage in raping trans men and trans women have the advantage in raping other women." People who train effective martial arts for several years have a similar advantage over the general population--should we ban martial arts training? No, because it turns out we have different system of incentives in place to prevent rape.
Third-"There are also genetics markers and brain area changes observable in people who develop schizophrenia. And also genetic markers and brain area changes in those people prone to be very religious..." I don't claim to be an expert in ANY way on brain physiology but this argument seems ridiculous if I read it as is. Wouldn't you know, human minds are governed by the physiology of their brains. There are endless arguments on nature vs. nurture, but from even a slightly deterministic view of personality development your argument can be used to call someone liking vanilla ice cream a psychosis. (I mean, I wouldn't necessarily disagree). If instead you are just pointing out that Mr. Alexander's argument proves too much and you just chose bad examples, I can see where you're coming from, but would point out that some people have the opposite problem of believing transgenderism to be entirely dependent upon upbringing/personal choice, and it is worth pointing out the science on the issue as it is. Full disclosure, I do not know if the science has changed at all in the past three years, but that has no direct bearing on the fallacies in your argument here. I will do some research on my own.
Fourth-"This is really the crux of the articles entire argument--we should do it to be nice." I didn't get that at all. The title of the article is "The Categories Were Made For Man, not Man For the Categories." It isn't "Be Nice to People." This is the issue with entertaining bad premises to point out how they're bad premises--a third party gets to cherry pick the argument and make an easy rebuttal. The question at hand is not whether it is nicer of cis people to accommodate trans people or of trans people to suffer in silence. The question is one of semantics. Now I'm going to do what I just said is a bad idea and say this--I am skeptical that the negative utility of accommodating reasonable requests of trans people (use preferred pronouns, for example) is greater than the positive utility of trans people on the receiving end of such tolerance. But more on the semantics point later.
Fifth- Accommodating trans-Napoleonism. See above, esp. including the "cherry-picking the easy argument to rebut" since you made no mention of the more robust objection in Mr. Alexander's article. I have nothing to add on this one until I hear such a rebuttal.
Sixth-Self identification. Some googling couldn't turn up the study you cite, but assuming it exists, then the issue would appear to, again, be largely regarding the prison system and not transgenderism.
One argument I thought was somewhere in between was the one comparing gender dysphoria with body integrity identity disorder. I'm tempted to dismiss this one because it seems like reducio ad absurdum, but its a fairly compelling argument. I could point to any number of other instances where we respect a patient's choice to effectively do themselves permanent harm. I believe by the same train of logic you would require a full psychiatric evaluation before any person ever got a tattoo, though that argument feels a little too absurd to be fair. There is a whole body of medical ethics on this and similar issues, but in my personal opinion it appears that the consequences of treating gender dysphoria with reassignment surgery are vastly different to the consequences of fulfilling a patient's desire to paralyze themselves. It is a very, very difficult thing to determine when a person is of sound mind and I pretty much agree that gender reassignment surgery should come along with an intensive psycholgical evaluation process. For similar ethical questions--is it moral for the bartender to let intoxicated people poison themselves further and pay him for the privilege? At what point is it allowable for an elderly cancer patient to make the decision to move to hospice care? These are difficult questions and do not lend themselves to universal answers. But if the doctor, medical ethics board, and patient in question are all in unity on the subject, I think it would be very difficult to argue either way.
Okay, that's all for bad-faith arguments I think. Now as far as my position on this matter and how it might differ from what I get from you--as I said, at the end of the day the argument you're putting forth is essentially semantic. Scratch that--part of it is semantic and part of it is misattributing societal flaws to the existence of trans people. (Can you put forth a actual reason why men, let alone trans women, shouldn't go in women's bathrooms that doesn't devolve into victim blaming? Did we start posting guards at bathroom doors to check peoples genitals when I wasn't looking?)
But the semantic bit. My point, and I think Mr. Alexander's point, is as follows: if you define "male" as a person with a Y chromosome and apply this rule unilateraly, you will produce a group of people who mostly share similar characteristics, with few enough outliers that it is still a good label. If you identify "man" as a person who identifies themselves as a man, you also produce a group with functionally the same degree of similarity. What makes one category better than the other? Well, if you ask me, you can have your cake and eat it too, since I just used two different words to describe two different, self-consistent groups. As a medical professional, I think of sex along biological terms and gender along self-identification terms, and have thus far managed to avoid asking a biological male if he is menstruating regularly. Language is sufficiently versatile and subject to change that the semantic portion of the argument that boils down to "should we change the meaning of a few words" appears to be a resounding "yes" for me. The obvious counter is that, if we keep changing the meaning of a few words for a relatively small group of people, eventually our language will become nonsensical. I won't point out that this hasn't seemed to be an issue for any of the other words that evolved in response to new social phenomena, but I will say that I would be much less in favor of my (and others') proposed changes if they weren't so obviously precise. And, in fact, I am in favor of adding words to the language to accommodate any person who needs accommodating, since the only way such changes would become widespread is if enough people are in need of accommodation that adding the word would appear to be a good idea after all.
Now then, I know you will probably never read this response, but I hope it provides some balance to anyone else reading this article in the future. (To the person I recommended this to about an hour ago, I didn't mean to write this when I did).