I think having a fast hold on believing evolution is all that created the world requires an amazing faith that most mutations have been good efficient and fast.
The speed of evolution is not as disputed as you suggest.
Even if you would believe in Noah's ark, you would have to believe in extremely quick mutations. All dog breeds we have today didn't exist back in those days, neither did horses, or chicken, or many other species. But on the other hand, there are many fossils of many different kinds of animals that do not exist today. They disappeared, and others appeared. No single point in time of all showing up.
Has anyone calculated the number of selected mutations just one organism required?
About every second human being has at least one unique mutation in their genes, statistically. Statistically, everyday when you go to the bathroom and do a #2, there are hundreds of different new mutated lifeforms in your excrement. (I think the source of that one was one of my anthropology teachers. LOL!)
There's been many experiments where the DNA has been sequenced and the new generations too, and the mutations have been recorded.
Currently, there are organizations you can send your blood to and they can trace your lineage based on the mutation markers in your DNA.
And on another note, if DNA didn't mutate, we all would have the same DNA. That would be a devastating blow to the court system and identifying suspects using DNA.