Does using peroxide help clean my mouth?

Mouth cleansing. Commercial mouthwashes would be a much better choice for you rather than peroxide for oral cleansing. Peroxide is not meant to be swallowed, which is a risk if you use it to clean the mouth.
Slightly. If you want a clean mouth, i recommend brushing the teeth, gums and inside the cheeks. Use a tongue scraper and floss your teeth daily. Followup with closys mouth rinse.