I1. Yes, but a "cleanser" itself can mean multiple things and people do not usually say "face cleanser."
3. Yes, but people usually only say shampoo.
4. Yes, but it sounds better to say "I used facial cleanser to wash my face" or "I use a facial wash to cleanse my face."
5. You can just say "I use shower gel to wash my body."
6. You can just say "I use shampoo to wash my hair" or "I wash my hair with shampoo."