There has been some evidence that gender might exist in some way in primates.
https://psmag.com/environment/do-animals-have-gender-roles
Obviously human social structures are radically different from other creatures. Since gender is a social construct, I wouldn't expect many other species to demonstrate it. You don't see zebras wearing dresses. We do a lot of things as humans that other animals don't. Many ways we show our gender aren't naturally demonstrated. Other animals don't dress up, or put on makeup, or shave, or communicate in as complex a way as humans.
I'm not sure why your second and third questions are relevant.