What did Sarah hope to achieve by giving Hagar to Avraham?
Rashi and Ramban #1: She hoped that as a reward her for her selflessness in giving her to Avraham to have children, in spite of the feelings of jealousy it would inevitably spark off, Hahem would allow her too, to conceive. 1
Ramban #2: Because of the tremendous satisfaction it would grant Avraham to have children, even if it was not from her.
Seforno: Perhaps the jealousy that Sarah was bound to feel when Hagar would give birth to a baby would arouse her physical functions, and she too, would conceive a baby.
Ohr Yechezkel (Midos, p. 231): Anyone who suffers pain and disgrace for Kevod Shamayim merits great success and Berachos.
Rashi writes: "I shall be built up through her - in the merit of bringing a rival wife into my home." Gur Aryeh explains: The intent of this verse is not that Sarah would deem Hagar's future children as if her own, but rather that this would be a source of merit for Sarah, for her to bear children herself. Sarah said, "May Hashem see how much I desire that my husband Avraham have children, and enable me to bear him children as well."
Why did Sarah see fit to mention that Hashem had stopped her from having children?
Seforno: Although Hashem had promised to give Eretz Yisrael to Avraham's children, He did not say that those children would be from her; so she gave him Hagar to help fulfill the prophecy. 1
How does the expression of "building" (Binyan) apply here?
Rashi #1: A woman who has no children is referred to as "Harus" (demolished, broken).
Rashi #2 (to Kesuvos 112a): "Building" is an expression of bearing fruit (whether it refers to land or to children).
Why does the Torah inform us that "Avraham listened to Sarah's voice," and not just, "and he did so"?
Ramban: To teach so that this is something that he would never have done without Sarah's consent, and that, even now, he only took Hagar because Sarah told him to and not because he wanted children (which of course he did)!
Seforno: He examined her words to see whether her offer was genuine, and it was only when he saw that it was that he took Hagar.