How should we deal with our father's second wife and her children, and is our father entitled to demand that our mother meet her?

The children of this woman are your brothers, and maintaining ties with brothers is obligatory because they are kin ( رحم - rahim), and severing ties of kinship is among the major sins. It is permissible for your father to ask your mother to meet his second wife, and it is even desirable for the sake of companionship ( ألفة - ulfah), but if she refuses, she cannot be forced. It is not permissible to house two co-wives together in one dwelling except with their consent, due to the animosity and disputes it may cause. Your father is not permitted to force your mother to visit her co-wife, and your mother is not permitted to prevent you from maintaining ties with your brothers from your father's side.