In this exercise, we are dealing with third conditional sentences. These are used to talk about hypothetical situations in the past, things that did not happen. The structure for the third conditional is:
If clause : "if" + past perfect (e.g., "if I had known")
Main clause : "would have" + past participle (e.g., "I would have gone")
Let's identify and correct the mistakes in each sentence:
Original: I couldn't stay with you. I wouldn't have left if I didn't have the appointment. Correction: I wouldn't have left if I hadn't had the appointment.
Original: My sister didn't want to marry Jim. If he had proposed to her, she would refuse. Correction: If he had proposed to her, she would have refused.
Original: We couldn't buy the house. But we would have done it if we had enough money. Correction: But we would have done it if we had had enough money.
Original: I didn't catch a word. If I had understood, I'd give them some advice. Correction: If I had understood, I would have given them some advice.
Original: If he wasn't the best player in the team, he wouldn't have earned so much. Correction: If he hadn't been the best player in the team, he wouldn't have earned so much.
Original: Peter was furious. He wouldn't lose his temper if he hadn't done all the work on his own. Correction: He wouldn't have lost his temper if he hadn't done all the work on his own.
Original: The kitten didn't want to leave the house. Otherwise, it would have played in the garden if it didn't rain cats and dogs. Correction: Otherwise, it would have played in the garden if it hadn't been raining cats and dogs.
Original: I didn't know that he was at home. I would drop in if I had known it. Correction: I would have dropped in if I had known it.
Original: They wouldn't have traveled by ship if the authorities didn't close the airport on the island. Correction: They wouldn't have traveled by ship if the authorities hadn't closed the airport on the island.
Original: She was so pretty. If I were twenty years younger, I would have asked her to marry me. This sentence is correct because it follows the correct conditional structure where "If I were" is the correct use of the subjunctive mood in the 'if' clause combined with "I would have asked" in the main clause.