4) Can you explain why gender identity does not exist? Don't cisgender people identify as/feel like they are the sex they were assigned at birth? Don't other animals identify as their sex assigned at birth due to lack of cognitive ability to identify as anything but their own sex?

I don't think anyone has done research on the existence vs. non-existence of gender identity. All I can say is that I don't have an abstract gender identity and when I read about its presumed universal existence I have no idea what it means and why others find it such an important concept. People have personalities, true, and those show some pattern of being distributed differently by sex, but most everybody really is non-binary in the sense that most people are not walking two-dimensional Barbie or GI Joe stereotypes. Stuff that I see as part of someone's personality is assigned to that person's gender now. This is bad for feminism as it makes the gender boxes more rigid.

Because I don't believe that most people possess an abstract gender identity, I am very reluctant to assume that if only animals were able to think on a higher level they, too, would somehow be found to have such an abstract identity. In any case, studying something like 'gender identity' is impossible, because it is supposed to be an internal feeling and others cannot validate it by measurements etc.

I believe that the concept makes sense for transgender people, however.