Is a wife obligated to serve her husband's ill mother, who can only move with difficulty? And is this considered an act of a husband's filial piety towards his parents, or an injustice to his wife? And is she obliged to obey him in this matter?

It is not obligatory for a wife to serve her husband's mother, and the husband has no right to compel her to do so; otherwise, he would be unjust to her. Honoring one's mother is required, but not by oppressing the wife and obliging her to do something that is not religiously incumbent upon her. If the husband's mother needs service, this is an obligation upon her children, each according to their capacity.