1. I was late for the meeting thanks to the rain.

2. I was late for the meeting owing to the rain.

(Can we use 'thanks to' in Sentence 1? And Sentence 2 has no problem, doesn't it?)

3. The boy went to the teacher's room to be disciplined by his homeroom teacher.

4. The boy went to the teachers' room to be disciplined by his homeroom teacher.

(Which one is correct?)

Sentences 1 and 2 are fine.

3 is fine if there's only one teacher who uses that room. If you're referring to a room that all the teachers use at one time or another, then 4 is correct.