There is considerable evidence of gender identity different from biological sex lining up with biological features in several areas more typical of the other sex, so it seems that gender identity is tied to biology.
Which, if you think about it, it has to be: humans are biological machines, everything about them is biology, on one level or another. All of psychology is, ultimately, biology.
As for the second, saying psychology is ultimately biology is reductionist in a way that dismisses the context and environment biology gets expressed in. You would be a much different person if the same set of genes was born in a different place, or at a different time, or if your life circumstances were different in any mild way.
Yes, it is, which would be relevant if someone suggested there was no physical explanation for a psychological phenomenon (which is, in fact, what they are usually claiming when they claim it is not biological.)
The bigger argument I get into with coworkers is the blurry line between brain function/behavior stuff and psychology.