I can think of one partial exception to #1, and that's if there's a child involved and if one or both parents are using FB as a place to put photos and news of the child. Staying in touch with an abusive ex is generally not healthy, but adding in a child makes it not about the relationship with the ex. Children under 13 aren't allowed to make their own accounts on FB (or other social media sites), even with parental permission, due to US laws.

If there's a child in case #2, that's more complicated, but as the girlfriend would be the other parent in that case, the decision really should be hers. Well, unless the abusive ex is a close relative of the guy's such as a sibling.