I really should have answered the trans* thing thoroughly when it was brought up, but was sleepy! Basically the asterisk is a wildcard to make it explicit that the word is for the entire trans spectrum.
Language here is gooey and really not well defined, so while I see it being basically interchangeable with "trans", some people see "trans*" as more inclusive and "trans" as less inclusive.
"gender dysmorphic disorder" is umm a pretty rare phrase (6 pages on google), and certainly isn't anything from the DSM. Could stem from mixing up the DSM "gender identity disorder" with "body dysmorphic disorder" and the phrase "gender dysphoria", but I don't really know.
So yeah, I mean they sound trans to me and it's likely that they have some gender dysphoria but I'm really not comfortable guessing and can't be sure or speak for them.
In any case, even if it was some official thing I'd recommend listening to them about their experiences and try not to assume stuff about them based off of what X medical diagnosis says. *shrug!* Sorry if that wasn't super helpful xD;
As for the word "genderqueer", way less comfortable speaking on this, and less sure, and all binary myself. So take this with a huuuuuge grain of salt (it is true! I cannot resist answering questions!). And again, language here is still very much in flux, but I see genderqueer as a very loose and broad sort of identity word for anyone who doesn't fit in the gender binary and wants to use it.
Different genderqueer people may have wildly different genders, or experience gender totally differently (though binary people could also experience gender totally different from eachother!). I don't think it implies much beyond that. Though if I were trying to use a catch all I'd probably go for "nonbinary people" (which uses language that centers the binary, so isn't perfect
, but seems like more of a catch all to me).
For fluid gender I tend to think more of "genderfluid", "bigender", and "trigender", though of course a genderqueer person could have a fluid gender, and someone who identifies as genderfluid could also identify as genderqueer, etc.
Again though I'm a little familiar with this, I am still totally speaking from a place of privilege here, and not super familiar, so yeah, keep that in mind.