What is the ruling on adding non-mahram female relatives on Facebook, and is one sinful if a woman's picture appears next to their name when commenting on their posts, and what is the ruling on having friends who publish forbidden things on their pages?

You must prevent any comment or "like" that displays an image of a woman showing what she is not permitted to show of her body, or that contains anything forbidden. If you fail to do so, while having the ability to prevent it, you are complicit in the sin. To avoid awkwardness with relatives, you can write a post to alert friends that you will unfriend anyone who posts on your page or shares a post with you that contains something religiously forbidden, and that the reason for this action is your fear of sin. As for forbidden things your friends post on their private pages, there is no sin upon you unless your interaction on your page leads visitors to your private page to access theirs.