1. Looking through the window, a plane was passing overhead.

Looking through the window, I saw a plane passing overhead.

( 2 ve khong cung chu ngu nen sai :v)

2. Realizing that it was late, I ran all the way home.

It's correct.

3. Having examined me, I was given a prescription by the doctor.

=> Having examined me, the doctor gave me a prescription.

=> Having been examined, I was given a prescription by the doctor.

4. Turning on the television, the news reader said there had been an explosion.

=> Turning on the television, I heard the newsreader said that there had been an explosion.

5. Dropping the shopping bags on the kitchen floor, Mrs Moorcroft sighed and sat down heavily

It's correct.

6. Having practiced the piano, I decided to call one of my friends

It's correct.