Answer:
Sure, here are the sentences with the relative pronouns omitted:
1. The man next door to me is a teacher.
2. The dress Rose bought from the mall doesn't fit her very well.
3. The bed I slept in wasn't comfortable.
4. Who is that girl I saw in the park?
5. I remember the day I moved into my apartment.
6. U Phyo San, the manager of our company, has left for Singapore.
7. The cat with a pushy tail belongs to my aunt.
8. The building destroyed in the storm has been rebuilt.
9. She purchased a car parked outside the house.