This is pretty much the perfect answer to your questions -- I would reply in more detail, but I would just be regurgitating points from this article.

http://slatestarcodex.com/2013/02/18/typical-mind-and-gender-identity/

In short:

• There are known problems where people's mental 'map' of their body doesn't match their physical body. Two are Phantom Limbs, and the 'opposite effect', body integrity identity disorder. This is where a healthy person with four healthy limbs feels like one of their limbs is “wrong” and has a strong desire to get it amputated). This disorder has a very non-psychiatric resolution: if patients are able to amputate the offending limb, they are perfectly happy and never have any further complaints (other mental health problems which involve the desire to remove body parts don't get resolved by fulfilling the wish -- the desire just gets transferred to a new part).

• The same logic that suggests that sufferers of BIID just have a mid-match between their mental body map and their physical body can apply to trans people. I think it's hard for people whose physical and mental gender match to understand what this would feel like. I certainly don't 'understand' it, on a visceral level. But it could be a case of 'fish not knowing what water is'. We're so used to our physical and mental genders matching, we don't even notice that they are different. Certainly trans people report a huge amount of distress from being in the 'wrong' body, just as BIID sufferers report distress from having 'wrong' limbs. I see no reason to disbelieve them.

• A follow up point is, in short, "so what if we classify being trans as a mental health problem?". It doesn't mean we shouldn't 'treat' it in the most effective way. Trans people report that sexual reassignment surgery removes all their distress. That's the kind of perfect cure that medical researchers dream of. If someone found a similar operation that could effectively cure schizophrenia... that's the kind of breakthrough that people win Nobel prizes for.