That's one way to cash out the distinction, but "gender" can also mean sex. This usage (at least in English) dates back to the 13th century.

At best, it's ambiguous. You can't really know what someone means by "gender" without context, such as the mentioning of the sex/gender distinction. "Gender" has become somewhat of an auto-acronym.

This if often claimed but there seems to be no evidence for it. If anything, all efforts to change humans using social techniques have failed. Take gay conversion therapy, for instance. It just doesn't work. There is no evidence for it. We just can't mold ourselves using social pressure -- because it's really down to biology.

If gender is social/cultural and there are infinite genders and it is a spectrum, what makes gender different from personality? How is that term useful in a way different from personality, and why should pronouns be altered for it?

But they aren’t, at least in our present culture decoupled, so people have a sense of whether the feel aligned in form and physiology with their gender identity. We’re going to find out in future decades as people experiment with gender identity and physical transformation how much this can be deconstructed, how much of what we think of as gender is an association of things that often went together in the past, how much is about social perception, how much is about your own feeling of inhabiting your body, and how much is a free space for you to author