Is the husband's statement to his wife on Friday, "If my wife doesn't do such-and-such, then by my mother's life, Saturday will not pass with her still under my marriage," considered an oath, and what is its expiation if it is?

It appears that a promise of divorce conditional on the non-performance of a specific act does not constitute a divorce merely by the promise. As for swearing by "my mother's mercy," there are three cases: First, if you intend God's mercy upon your mother in this world, then this is permissible. Second, if you intend God's mercy upon her in the Barzakh (interim period between death and resurrection), this is not permissible because it is a conjecture about the unseen. Third, if you intend the mercy of a created being, this is not permissible because an oath can only be sworn by God. If, in your statement "and by my mother's mercy," you intend God's mercy, then it is considered a legitimate oath that requires expiation if you break it.