I have trans and genderqueer friends (one of them is an ex-JW, our mothers go to the same KH) and they're lovely people, and definitely show no signs of "mental illness". They just *know* what they are, the same way you just *know* you are male (and don't necessarily even "feel" male, you just know what you are). It was really painful for them to grow up knowing who they were, and being treated as someone else for so many years, and just as painful is the ignorant accusation of being "mentally ill" or "sick", which they get probably more often than you can imagine.
Gender/sexuality don't just fit into little black-and-white narrow boxes anymore. The difference between cutting off your hands and undergoing gender reassignment surgery? Well, for one thing, you're not left with a permanent disability. It's mainly cosmetic, to reconcile the physical body with your mental perception of yourself. Lots of people do that with plastic surgery - some subtle, some more over-the-top, but whether the results are to everyone else's taste is beside the point - it should only matter whether the person undergoing the surgery is happy.
Sure, there are inital side effects (as with any surgery), but they go on to lead normal, productive lives - happier ones than before the surgery, almost always. They still have sexual function, and of course, overall - it just makes them feel happy and right. They're consenting adults, and shouldn't be viewed as though there's "something wrong with them" in need of "fixing" - just like a gay person shouldn't have to be subjected to that "reparative therapy" crap. I'm sure you can find a couple of anomalies, as with everything, like the (yup, anecdotal) one you found on a message board. But for the most part, efforts to "fix" or "change" them are either futile or else do extensive harm.
It's also worth noting that it's not as simple as going to a doc and being like, "Chop off my penis, doc! Doooooooo it! DO IT NOW!" (By the way, I'm a bit fuzzy on this, but if I understand correctly, I've been told that they don't actually chop it off. They more turn it inside-out or something, to construct a vaginal-type opening that fits the cosmetic goal and also retains sexual sensation). They have to go to incredible lengths not only to prove that this is what they want, but that this is who they ARE. They have to be able to show that they've been living as whatever gender they consider themselves for a significant period of time. They have to undergo a barrage of tests and therapy sessions to make sure they really have Gender Identity Dysphoria. They have to do a ton of pre-surgery counseling to prepare them for what they're signing up for and make sure they understand completely what's involved.
It's nothing like going to a doctor and asking them to make you an amputee. People with Body Integrity Disorder can usually (though not always) be treated with therapy and medication. Gender Identity Disorder can't, or almost never can. My trans and genderqueer friends are great people and deserve to be treated as such.
Some genderqueer/trans people feel the need for the reassignment surgery. Others don't. They run the spectrum. It's the same with asexuality. There's a bajillion asexuals out there, and they run the gamut. Some asexuals still desire romantic-yet-celibate relationships with long-term companions. Some date other asexuals, some date non-asexuals. Some are willing to "go along with sex" for their non-asexual partner's sake even if it does nothing for them (to them, sex is sort of like writing out a grocery list or doing their taxes). Others find sex completely repulsive/squicky and want nothing to do with it. Some asexuals who find sex squicky have open relationships with their romantic-celibate partners (they are fine with their partner getting their sexual needs met by other people, as long as they come home to them). Some don't desire a romantic relationship at all. All are equally valid, none of them need "fixing". I'm sorry that you feel being asexual means there might be something "wrong" with you. Embrace your lack-of-sexuality - it's just who you are, and you're in good company :)
But lay off trans people. They get these hurtful questions all the time and I can't imagine having to live the way they do, having to constantly defend who they are against accusations of being "sick in the head". I get that a lot of it is just borne out of curiosity and a lack of understanding, rather than overt hostility (though they get that too), but it's undeserved and it can cause them a lot of anguish.