Is it permissible for the mother and sister to seize the questioner's apartment and prevent her from disposing of it, despite her dire need for money, and while they are renting out their own apartments at high prices?

You are not religiously obligated to house your mother or sister if they are self-sufficient; rather, doing so falls under good deeds and kindness. There is no objection to selling or renting the apartment. However, if the mother and sister knew that their use of the apartment was due to embarrassment rather than sincere willingness, then such use is impermissible for them; indeed, it would be ill-gotten gain.