1. Have you ever been / Were you ever to Australia?
2. His family went / has gone to Australia last summer.
3. Shakespeare wrote / has written many great literary works.
4. My dad is a writer. He wrote / has written quite a few detectives.
5. I like cycling. I took / have taken part in several cycling competitions this year.
6. In June Mike and Mary took / have taken part in a teenage club meeting.
7. Several days ago I met / have met my best friend. We were / have been friends since we were / have been children.

Правильные варианты такие:

1. Have you ever been to Australia?
   Здесь спрашивают о жизненном опыте: «Ты когда-нибудь был в Австралии?» Поэтому нужен Present Perfect: have been.
   Were you ever to Australia? — так обычно не говорят.

2. His family went to Australia last summer.
   Есть точное указание на прошлое время: last summer. Поэтому используется Past Simple: went.

3. Shakespeare wrote many great literary works.
   Шекспир жил в прошлом, его жизнь закончена, поэтому нужен Past Simple: wrote.

4. My dad is a writer. He has written quite a few detective stories.
   Отец жив и продолжает быть писателем, речь идёт о его опыте/результате к настоящему моменту. Поэтому лучше Present Perfect: has written.
   Лучше сказать не detectives, а detective stories или detective novels.

5. I like cycling. I have taken part in several cycling competitions this year.
   This year может означать период, который ещё не закончился. Поэтому используется Present Perfect: have taken part.

6. In June Mike and Mary took part in a teenage club meeting.
   In June здесь воспринимается как конкретное время в прошлом, поэтому нужен Past Simple: took part.

7. Several days ago I met my best friend. We have been friends since we were children.
   Several days ago — точный момент в прошлом, значит met.
   We have been friends — дружба началась в прошлом и продолжается сейчас.
   После since обычно используется Past Simple: since we were children.

Итоговый вариант:

1. Have you ever been to Australia?
2. His family went to Australia last summer.
3. Shakespeare wrote many great literary works.
4. My dad is a writer. He has written quite a few detective stories.
5. I like cycling. I have taken part in several cycling competitions this year.
6. In June Mike and Mary took part in a teenage club meeting.
7. Several days ago I met my best friend. We have been friends since we were children.