What should I do regarding my mother and siblings who watch a series full of suggestive content, and am I accountable for their sin? And what is the ruling on avoiding mixed-gender family gatherings for fear of sin and cutting off ties of kinship?

Since you have repented from watching TV series and advised your family to stop watching them, there is no blame upon you, Allah willing, as Allah Almighty says: "O you who have believed, upon you is [responsibility for] yourselves. Those who have gone astray will not harm you when you have been guided."

As for your sitting with the family when women are mixing with non-mahram men, if the mixing is suspicious or involves forbidden matters, then it is not permissible for you to sit with them. However, if the gathering is controlled by the regulations of Sharia, there is no harm in it.