Today's post (or yesterday's by the time I finish):
This one is even more ugly than usual for this topic.
(Before going on to dissect it, I should start with a disclaimer: as a straight cis male, my discussion of the subject is informed only on an intellectual level and not from experience. It's not my intention to offend anyone unnecessarily.)
Horn's first "question" (these are clearly not questions he actually wishes answers to) starts out by prejudging the entire case: the dichotomy between facts and feelings is a false one, introduced simply to dismiss feelings regarding gender identity as "not facts".
I have no idea of what's with the "transabled" thing (not qualified or sufficiently informed to comment on it), but one thing I know of that is significant is that there is a condition called "body integrity identity disorder". Scott Alexander (who as an actual working shrink is more qualified than I to comment on it) discusses it here along with some speculation on its possible significance for gender identity.
The second "question" can be answered very simply: "man" and "woman" denote two clusters of not-perfectly-correlated inferences about a number of biological, psychological and social issues. The most common way to break down the edge cases is to separate the concepts of biological sex (which itself is more than one variable, it's not just a matter of karyotype), gender identity, and gender performance, acknowledging that even these do not break down into sharp categories.
The third "question" is drawing a (particularly offensive) false equivalency between personal partner preferences and actual discrimination. Whether you find someone attractive or not is not the same thing as bigotry; you can find someone unattractive without needing to dehumanize them or discriminate against them.
The fourth question (which might actually be worth taking seriously as a question!) really boils down to how far we trust parents to make medical decisions for their children, and how much protection the children need from this. Children are not the property of their parents, and demonstrably parents sometimes act against the child's legitimate interests or rights.
Certainly it is inappropriate to perform surgical interventions or other significant treatment on transgender children who are not old enough to give adequate informed consent. But delaying puberty, when this can be done safely, is certainly an option that ought to be available under appropriate safeguards. The idea that young children are somehow going to end up getting reassignment surgery over a whim is pure scaremongering fantasy.
It's worth noting that McHugh is not a reliable source; and we're about to see that demonstrated:
The final "question" quotes McHugh in an attempt to claim that transgender identities are "harmful". But to do this, McHugh compares the suicide rate between transgender patients who had had surgical reassignment and controls from the cisgender population. Clearly this argument is bogus; even the authors of the study McHugh is citing specifically say that their data cannot be interpreted this way:
It is therefore important to note that the current study is only informative with respect to transsexuals persons health after sex reassignment; no inferences can be drawn as to the effectiveness of sex reassignment as a treatment for transsexualism. In other words, the results should not be interpreted such as sex reassignment per se increases morbidity and mortality. Things might have been even worse without sex reassignment. As an analogy, similar studies have found increased somatic morbidity, suicide rate, and overall mortality for patients treated for bipolar disorder and schizophrenia.,  This is important information, but it does not follow that mood stabilizing treatment or antipsychotic treatment is the culprit.
Horn then shows off his own ability to ignore inconvenient facts; it is objectively true that transgender people face substantial harassment at school or work or by law enforcement, discrimination in access to jobs or housing or even healthcare, and increased rates of violent assault or murder compared to the cisgender population. To brush this off as "framing" the issue—even though in spite of all this the proportion of patients who regret transitioning is tiny—is egregiously dishonest.