A transgender person has a mental gender that does not match the gender assigned to them at birth. Transgender people exist in both real life and in fanfiction.
Badfic often includes plenty of trans!fail, usually involving the idea that the character is "really" their assigned gender, rather than their mental gender.
- The mental gender, not the assigned gender, is the person's true gender. That means (if you think about it) that they are not actually "male trapped in a female body"—they are male; so their body is male even though it is atypical for a male (or vice versa).
- Grammatically, pronouns that match the mental gender make the most sense, whether or not they have made any physical modifications.
- Having someone find out the person they are dating is trans, recoiling in horror, and having the story playing it as funny? Not cool. Not cool at all.
- The opposite of trans is "cis". No need to make up any more words than we have to.
- Transman/transwoman work well if you have to specify, but there's usually no reason not to just refer to them as simply "man" or "woman". Some trans people like to use the acronyms FTM and AFAB (female to male; assigned female at birth) or MTF and AMAB (male to female; assigned male at birth) to make the distinction.
This is another phenomenon that falls under the trans umbrella that exists both in fan fiction and in real life. There are lots of variations, especially with magic and sci-fi technology involved, but genderqueer people are simply those who don't fit into the usual male/female categories. They may be in-between (intersex or bigender), switch periodically from one gender to the other (genderfluid), or simply not have a gender (agender). They may be third-gendered. In fiction, entire species of shapeshifters may not even have a concept of gender.
"Genderqueer" refers mostly to people whose species has a relatively firmly-established gender binary that they don't fit into—while shapeshifters or non-gendered robots may be neither male nor female, and be effectively genderqueer, they are not usually called genderqueer.
Unfortunately, English has no good firmly-established gender-neutral pronouns. Those wishing to read more on the matter can start with Wikipedia's article on gender-neutral pronouns.