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Yes, gender is derivative of psychological sex (or what is often called gender identity, though that is a confusing label because it's not quite like other kinds of identity).
It is certainly not a social fabrication though. People actually, physically, feel cool while under the shadow of a tree.
> and at worst an unmoored fantasy.
This statement is beyond the pale. You should probably retract it.
> Pronouns refer to sex.
This statement is pure, unadulterated gender ideology.
> Stop coercing my language and do what you want with your body.
Some amount of coercion is required to exist in a society—believing otherwise is right-libertarian nonsense. For example, in any reasonable workplace or school setting you are required to not harass, discriminate or retaliate against people. That is a kind of coercion. Part of that coercion is to use trans people's chosen names and pronouns. This is just. In a sane jurisdiction like California it is the law.
Outside of work or school you can do whatever you like.
No, gender is a product of society, it is a socially ascribed trait, within a socially constructed system.
Gender identity is a product of internal biological (including paychological, which is an epiphenomenon of biology) reality interacting with social context including social gender constructs. Calling it psychological sex is odd but saying that gender is a derivative of it is wrong (its true that there is an emerging belief that gender should be ascribed on the basis of it, but that's different.)
> It is certainly not a social fabrication though.
Gender, including the somewhat popular binary construction of it, is very much a social construct. “Fabrication” is not inaccurate, either.