How can the Prophet’s — may Allah bless him and grant him peace — prohibition against a believer’s aversion to another believer be reconciled with his divorce of Hafsa — may Allah be pleased with her — and his command to a man to divorce his wife, and what is mentioned about its implication in the slaughter of excess sheep?

There is no contradiction between the hadiths concerning divorce. Divorce was legislated by Allah for good reasons, as companionship sometimes becomes impossible. The hadith: "No believing man should hate a believing woman" is an invitation to pause and reflect. If a husband dislikes a trait in his wife, he may find another trait pleasing. He should be patient with her, for Allah Almighty says: "But perhaps you hate a thing and Allah makes therein much good." The Prophet's—peace and blessings be upon him—divorce of Hafsah and then his taking her back by the command of Gabriel because she was "a frequent faster and a frequent worshiper at night" does not contradict what has been stated. Some people are commanded to divorce for an overriding benefit. In summary: patience, reflection, and striving to preserve the family whenever possible. If it becomes impossible, then divorce is a relief for both parties.