Sure, here are the sentences converted into negative sentences, yes/no questions, and their corresponding answers:

1. Mrs. Nguyet teaches Maths.
- Negative: Mrs. Nguyet does not teach Maths.
- Yes/No Question: Does Mrs. Nguyet teach Maths?
- Answer: Yes, she does. / No, she doesn't.

2. My brother goes to the library twice a week.
- Negative: My brother does not go to the library twice a week.
- Yes/No Question: Does my brother go to the library twice a week?
- Answer: Yes, he does. / No, he doesn't.

3. I do exercise every morning.
- Negative: I do not exercise every morning.
- Yes/No Question: Do you exercise every morning?
- Answer: Yes, I do. / No, I don't.

4. He should stay in bed.
- Negative: He should not stay in bed.
- Yes/No Question: Should he stay in bed?
- Answer: Yes, he should. / No, he shouldn't.