As I said in the beginning of the thread, you will never be able to do advanced clothing with thickness, belt and borders, etc etc without loop selections and the help you get from a nice clean quad mesh. It is not enough to combine two triangles into a quad it must be orderly.

"The mesh to the right was changed, it was decimated, simplified"

I you are going to work further with it keep it simple (KISS). In the end you can choose any level of subdivision you want and manipulate with magnets or brush or whatever. Are you working with conforming clothing?

"PropViewer doesn't change the number of vertices and its position"

So sad, because that is what I want.

Finaliy, UV map.

It is better to work piece by piece instead of taking on the whole mesh at once. An good work flow is to make selections of all the UV islands, and then retopoligize them one by one. So the UV maps are maintained.