1. Her mother got out of the room.

2. Her mother went out of the room.
3. Her mother walked out of the room.

4. Her mother got into the room.
5. Her mother went into the room.
6. Her mother walked into the room.

7. Her mother got in the room.
8. Her mother went in the room.
9. Her mother walked in the room.
[Are they all grammatical? Can we use both 'into' and 'in' in this case? Which ones are commonly used?]

They are all correct. "got into" implies there might have been some difficulty, as if she had to crawl through a window or break down the door. "I had to break a window, but I got into the burning house to rescue the dog." "walked into" or "went in" implies normal circumstances.