Good thing thats not whats happening
Yes, that's what is happening.
When you say "a trans woman is a man", you are negating the woman identity of said person and its trans identity as well. You are therefore negating her existence.
There is no way around transphobia. You can find all the excuses you want, it won't work.
Yes they do both.
(you have a really hard time on this topic, its really problematic mate)
Yes, they do both. They are precising the sex of the baby AND they are also telling what gender the baby is ("its a boy", "its a girl").
Remember, sex and gender are two very different things, so as a doctor by pushing a gender on a person without actually knowing what gender said person will identify as in potentially harmfull and will be the starting point for gender disphoria in the future.
In some cases even, some doctors will see that the sexual characteristic are not clear and both parent and doctor will PUSH a gender on what we call an intersex baby and BUTCHER THEM to make them correspond to the expected gender. This is how some person are butchered for life because of early operations. Why ? Because society does not recognize intersex people as legitimate.
What doctors should do, is ONLY precise the sexual characteristics of a baby. And that should be followed by as MINIMUM gender affirmation as possible until the child grows old enough to starts actually thinking about their gender identity. At this point, said child will be free to explore their gender without the carcan of heteronormativity and we will most likely get rid of the most prime causes of gender disphoria. This will result in children with more freedom and adults with LESS prejudices against transpeople.