As to the contaminants that are in you water heater... um, where do you suppose they came from? Those contaminants are in/from your tap water. So it doesn't really matter if you use the hot or not. They both have about the same amount of crud in them.
Also, it doesn't matter if you have city or well water either. In fact, unlike 50 years ago, you are better off with city water in many parts of the country. Here in Florida, over 40% of the wells tested contain choleraform bacteria that exceed 'acceptable' levels. Over 50% contain Nitrates.
The only solution is ion-exchange purification or R/O. There is no such thing as a filter that can remove some of the toxins that are in today's water. As to treating it chemically.... well, are you really removing anything? Or are you just adding another chemical to the mix that's already in there? The contaminants are still there.