More than being the first it means to assume the leadership position on something. Someone else might start something, for example, but you can still take the lead in making it happen. It implies a certain responsibility or initiative.

Yes, but more specifically it means that other people will also do it, second, third, etc. So if he is the only person doing it, then you cannot really say that he takes the lead. In the case of a conversation, there are two people, one will speak first, then the other will speak second, so you could say that "he took the lead in talking WITH her".