Which isn't to say that mutations don't play a large role in other cells. The average child has at least one mutation but usually they don't do much. Well, doesn't do much or means the zygote is totally nonviable and so never results in a pregnancy. Having children when you're older will tend to increase the number of mutation, especially for men where their gametes are produced throughout their lives. A cell from, say, someone's skin will tend to have even more mutations but that's a quantitative thing rather than a qualitative one.
Do you have a source for this? How would a neuron with a mutation be any different from the outside in terms of its axons?
No, I think cell differentiation is a completely different topic -- gene expression/inhibition and cell specialization has nothing to do with random errors to DNA, as far as I understand. So the problem is not actually solved at all. Especially since we are not usually trying to clone babies, whose stem cells would have incurred less damage.