Is it true that Sarah hated Hagar, and that Abraham (peace be upon him) divorced Sarah because of that? And was Sarah pregnant with Isaac or had she given birth to him when Abraham divorced her? And is Ishmael three years older than Isaac?

The jealousy between Sarah and Hagar after Ishmael's birth was the reason Abraham took Hagar and her son to Mecca. This was mentioned by scholars like Ibn al-Qayyim and al-Qurtubi, and it was by divine revelation from Allah Almighty to alleviate Sarah's jealousy. As for Abraham divorcing Sarah, there is no statement from anyone to that effect. What Abraham said to Ishmael's first wife was, "Change the threshold of your door," and it was not addressed to Sarah. His father Abraham commanded him to divorce her and marry another.