Verified answer

1. Jane read a book after she had done her home exercises.

  Jane read a book after doing her home exercises.

or

 Jane read a book having done her home exercises.

2. Dan couldn’t enter the University as he had failed the entrance exams.

  Dan couldn’t enter the University having failed the entrance exams.

3. After Nick had moved to Italy, he spoke Italian well.

 After moving to Italy Nick spoke Italian well.

  or

  Having moved to Italy, Nick spoke Italian well.