Is shunning a father who does not pray and drinks alcohol, in order to spare one's wife from witnessing his actions, considered a severance of kinship ties that incurs sin? And what is the correct course of action in this situation?

It is impermissible to abandon one's father for any reason, whether he be a Muslim or a disbeliever. This is supported by the Almighty’s saying: "But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them, but accompany them in [this] world with appropriate kindness." If the father abandons prayer or commits forbidden acts, the duty is to advise him gently, and to accompany him with kindness without assisting him in his disobedience, and to leave him when he commits the disobedience. It is permissible to visit him at times when he is not committing immoral acts, or to visit him alone if it is difficult to bring your wife along, because maintaining ties with your father is obligatory upon you, not upon your wife.