Is it Islamically permissible for a wife to refuse to host her husband's son from another wife in the marital home, and to prevent him from staying overnight or traveling with his father, claiming that Islamic law does not obligate her to do so? And is this considered a violation of the husband's qawamah (guardianship/authority), a severance of kin ties, and an injustice to the children?

A wife has the right to an independent house with its facilities and entrance. She also has the right to prevent her husband's relatives from living in it, unless the house is large and she has an independent section within it, in which case there is no harm to her from the husband's relative residing in the rest of the house. This right pertains to preventing residence and permanent settlement, not to preventing visits. A resident is one who settles permanently, while a visitor is not considered a resident. Therefore, the wife does not have the right to prevent her husband's son from visiting his father, even if it requires him to stay overnight for a day or two. Both spouses are advised to be tolerant and fulfill their respective rights.