Is it obligatory to obey and be dutiful and kind to one's father, and to treat him well, even if he is of bad character and irreligious, and mistreats his children and their mother?

If your father is as you have described him, then he is an oppressor and an abuser, especially if he insults religion and abandons prayer. In this case, it is permissible to detest him. However, it is not permissible for you to disobey him or cut ties with him. Rather, you must accompany him with kindness and patience, as Allah Almighty commanded, even with polytheistic parents. You must strive to enjoin good upon him and forbid evil from him with wisdom and gentleness, and work towards his rectification, and supplicate for his guidance.