Maybe not as much as you'd think. As an example, the EDAR Gene (https://www.thecrimson.com/article/2013/2/21/sweat-gene-iden...) affects, among other things, hair thickness and sweat gland density. It's isn't a small effect, and yet it's a single gene change in an otherwise highly conserved gene pathway.

Yes - hominims lost their fur before homo neanderthalis/homo sapiens. (Also it's probably more than a single mutation - more like a series of them.)