Philosophical Questions about Parshat Vayera

 Lot is saved because Hashem had mercy on Avraham, that Avraham would be too forlorn about the death of Lot that it would cause undue pain to Avraham. As for Sarah Imenu, who dies after hearing that Yitzhak had been bound, why was mercy on Avraham not shown in that instance? Certainly, more mercy should have been forthcoming towards Avraham vis a vis the pain he would experience on learning of Sarah's death than Lot's. 

Hashem, who allows his name to be erased in the case of a woman accused of licentiousness, does so to preserve shalom bayit, peace in the home. If that is the case, why is it that Hashem tells Avraham that Sarah laughed, setting up a showdown between them, wherein Sarah denies having laughed and Avraham reasserts that she had done so? 

A brief nice idea I heard from Rabbi Adiel Levi, with whom I teach at Yeshivat Maarava, is that when Avraham brought the supplies needed for the binding and slaughter of Isaac, though Avraham had Isaac carry the wood, he could not, in his heart of hearts have Isaac carry the "maachelet" the knife that would have been used for the slaughter itself. This shows Avraham's humanity in the face of what would seem like an inhumane, almost God-like act that abnegates any vestige of human sentience and feeling.