What is the ruling on staying away from family members who are religiously open-minded and who mock committed Muslims, in order to avoid their actions and minimize friction with them? Is this considered a severing of kinship ties?

Adhering to religion is not extremism (tashaddud); rather, extremism is exaggeration in carrying out a command in a way that burdens oneself. Whoever adheres to their religion and follows the Sunnah of their Prophet is not an extremist. If family members mock, this is a tribulation that requires patience and wisdom in dealing with them, and gently teaching them the truth. Whoever fears fitna (temptation/strife) in their religion because of them should distance themselves from them to the extent that preserves their religion. If they are merely harmed by them, then they may keep a distance that avoids the harm, but they should not abandon them completely.