Ответ:

  1. He sold his house in the city and bought a new one by the sea. Buy has the irregular past form bought.
  2. We played video games and then we went to the party. The two completed actions are in the past simple.
  3. I felt very excited while we were listening to the concert. The past continuous describes the ongoing action.
  4. They were standing at the cooker while they were making dinner. Both actions were happening at the same time.
  5. It was raining when we went to the park. The past continuous describes the ongoing weather; the past simple describes the event.
  6. I went out just after you arrived. The actions happened successively, so the past simple is used.