Is a will to donate half the value of an apartment to Allah—in the absence of children—to prevent the testator's paternal half-siblings from inheriting, considered legally forbidden (haram)? And what is the correct lawful action in this case?

If your mother waived the debt she owed, the debt is nullified, and the apartment becomes your property, with your mother having no right to it. The jurisprudential rule states that "what is relinquished does not return." So, once a creditor relinquishes their right, it is nullified, and they cannot claim it again. As for your mother's will to you, stipulating half of the apartment [should go to someone else] after your death if you have no children, you are not obligated to obey her in this; because a will intended to disinherit heirs is contrary to Islamic law. A lawful will is for a non-heir and should not exceed one-third [of the estate]. You must be dutiful to your mother, treat her kindly, and be gentle with her, without violating Islamic law.