1. She said that she liked animals.
2. He exclaimed that summer starts in June.Здесь нет правила согласования времен,т.к.это истина.Лето всегда начинается в июне.
3. Jane knew that her relatives would come.
4. She said me today that she still didn't believe me.
5. He said that if he had had some money he would have invited me for lunch.