Is it permissible for a woman have a haircut where they shave half of her hair off and leave the other side of her hair untouched - a style strongly associated with lesbians, bisexuals, transgender men, queers?
I really can't see an issue here. There's a discussion of a woman getting "a man's haircut", but that's not what this is.
There is a Talmudic concept of chashad -- don't do things that will likely cause an average person to think you're violating the Torah, even though you aren't. One example is a house that faces two different streets -- back when everyone's Chanukah menorah was visible from the street, we advised lighting two, so that the people on Main Street wouldn't think you're a bad Jew who failed to light for Chanukah (when in fact you did light facing Pleasant Road). There's a balance between that concept and the obligation to give people the benefit of the doubt; some even suggested that you give people the benefit of the doubt, but not houses. Long story short -- if it's truly likely that many people will seriously think this woman is violating the Torah just by looking at her haircut ... okay maybe ... but I don't think that's the case in normal circumstances. (I also won't get into the whole issue of how silly and mistaken some stereotypes are.)
Rambam, Prohibited Relations 21:9 reads (emphasis added):
ויש לאיש להקפיד על אשתו בדבר זה, ולמנוע הנשים הידועות בכך מלהיכנס לה ומלצאת היא אליהן.
As female-female relations are prohibited, a man should object if his wife wants to "go out" with women who are known to engage in such behavior.
I think the key word there is "known"; i.e. don't pretend the obvious isn't happening. That's very different than "worry about people drawing silly conclusions that you're not keeping halacha when in fact you are."