Question

1. Saying hey, the foreigner walked toward me.

2. The foreigner walked toward me, saying hey.

3. The foreigner, saying hey, walked toward me.
(Are they all grammatical?)

4. When I was given the mug, I felt happy.
4-2. Given the mug, I felt happy.

5. Because it was written in English, the book is hard to read.
5-2. (Having been) Written in English, the book is hard to read.

6. After the window was opened, the boy came in.
6-2. The window being opened, the boy came in.
6-3. The window opened, the boy came in.
(Are they all grammatical? Can we omit 'being'?)

7. As the door had been opened, the boys could enter the room.
7-2. The door having been opened, the boys could enter the room.
7-3. The door opened, the boys could
enter the room. (Are they all grammatical? Can we omit 'having been'?)

Answer 1

4-2, 6-3, and 7-3 are incorrect. Don't use them.

5-2, 6-2, and 7-2 are all awkward. It's best not to use them, but not errors if you do.

1, 2, 3, 4, 5, 6, and 7 are all correct.