Hello, teachers!

Are these two options [him & his] both correct and natural?

1. She felt him/his touching her back.
2. I heard him/his crying in the bathroom.
3. I noticed him/his watching me.
4. She saw him/his watering the flowerpots.
5. She watched him/his using chopsticks and corrected it.
6. I've heard of him/his teaching math at my alma mater.
7. She listened to him/his explaining the rules.

Thank you very much.
Enjoy the chilly wind.
Only the versions with "him" are convincing. To my ear, Sentence 6 is a bit awkward either way.
[I've / I] heard that he [taught / was teaching] math at my alma mater.
Hi everyone,

I'd love to hear about your ideas on the following sentences.
She felt him touching her back/she felt his touch on her back.
She listened to him explaining the rules/She listened to his explaination of the rules.
 CalifJim's reply was promoted to an answer.