What if I was to use bleach or Mr. Clean to rinse my mouth? Would it cause cavities or damage my teeth in some other way?

If it didn't damage your teeth, it would hurt your mouth. And if you accidentally swallowed any of it at all you could become very ill. A cheaper, safer way to whiten teeth without buying whitening products or going to the dentists is baking soda. You probably won't see immediate results but over time it will whiten your teeth without causing damage or illness.

NO!
Saliva contains necessary components, and bleach will kill them. Many people argue that even mouthwash (alcohol and flavoring) is bad for you. Although 98% water, saliva also contains electrolytes, mucus, antibacterial compounds, and various enzymes. The enzymes in the saliva break down some of the starch and fat in food as you chew, and also food caught between teeth, where it helps protect from bacteria attacking the food and teeth.

Whitening strips do contain bleaching agents but they stay on the tooth and in the strip.