Should I say ‘him’ or ‘his name’ in the given sentence? Are ‘correct,’ ‘which,’ and ‘had given’ correctly used?

Should I say ‘him’ or ‘his name’ in the given sentence? Are ‘correct,’ ‘which,’ and ‘had given’ correctly used?

1. Teacher Henry wrote his full name on the chalkboard so that the students could easily remember (him, his name).
2. Our professor was quick enough to correct certain responses from us students, which he had found erroneous or questionable.
3. “You must have asked follow-up questions if your students had given vague responses,” the principal told Mrs. White.