1. Typically, but not always. Some trans women are biologically intersex. (This also depends on how you define "biologically male" which is not totally straightforward.)
2. It matters in some contexts, not in others. Their physician should know, because various hormone treatments cause different effects in people's bodies, and certain health conditions effect biologically male or female people differently too. That's nobody else's business but the patient and their trusted medical providers. As far as their dignity, opportunities, and general acceptance, it doesn't matter. Trans folks deserve the exact same rights, opportunities, and acceptance as anybody else.
3. Usually people who bring this up aren't acting in good faith, so I don't engage with them. On the rare occasion where somebody is genuinely curious and wants to learn, I answer them in the same way as I am doing right now.
4. Because the word "woman" denotes multiple concepts, like the word "parent". If a child is adopted at birth and is raised by a couple, the child and their community will refer to those people as the child's parents. This is not a false statement, because the word "parent" doesn't only mean the direct biological progenitors of a person. Parent also is a social role, hence the verb form "to parent somebody." This is also why we have the terms, "biological parent" and "adoptive parent" to add additional information when it's necessary.

Trans women are women in the sense that they are filling their society's sociological role that surrounds the expected concept of a woman. That will be different depending on many factors, and will have many different aspects including their pronouns, fashion and clothing, voice, makeup, hair, activities, and so forth.

Just like any other woman, they will chose which social roles they desire to fit into, and which ones they don't, and all of that is completely acceptable.