Is it possible for women to recite at the Seder some parts of the Hagadah in front of other people, including men (I’m refering to a family Seder, when all the men are married or young boys)?
Reciting the Haggada so that others who hear the Haggada fulfill their obligation of telling the story of our redemption from Egypt is based on the principle "shomea k'oneh" - hearing something said from someone commanded to say it is like saying oneself that which is commanded (see Shu"t Chatam Sofer Orach Chayyim 15). The person reciting the Haggada must intend to enable through his recital all those who hear him to fulfill their obligation. Because it is not clear that all those reciting can maintain this kavana=intention, some poskim recommend that all those present recite the haggada quietly to themselves even as one person -in some seders there is a rotation - recites it out loud. Because one talmudic opinion views a woman's obligation to recite the haggada as rabbinic while a man's obligation is mandated by the Torah (see Pri Megadim Eshel Avraham 479 2), if a woman is the one who recites part of the Haggada out loud it is required that the men say this part to themselves quietly. There is no problem of Tzniut with a woman reciting part of the Haggada (this is comparable to her giving a dvar Torah at the table) unless she is singing it - in which case those restrictions regarding women's singing in mixed audiences apply.