Is it better for a woman to perform the Tarawih prayer at home or in the mosque, and is a woman sinful if she does not inform the congregants about the presence of a prostration of recitation during the prayer?

The basic principle is that a woman's prayer in her home is better than her prayer in the mosque. However, if her prayer in the mosque is longer and more devout, she is permitted to go out, provided she adheres to the Islamic guidelines. As for not alerting the women to the prostration of recitation (Sajdat al-Tilawah), there is no sin upon you for that, because speaking invalidates the prayer. You could have raised your voice with the Takbir or the supplication of prostration to alert them. Their prayers are valid because they did not intentionally add anything, and the Imam bears the responsibility for the forgetfulness of those praying behind him.