[KenH]

The little grey rectangle shows that the quad surface is not flat. i.e., that the four vertices are not all lying on the same plane. You would either have to Surface>Flatten, which could mess up surrounding geometry, or triangulate the rectangle. But the latter would create problems if subsequently subdividing.

The only method I can think of for your second question is to Surface>Cut-Away Object for the part you are working on and locking the rest of the object, then merge back together again and weld vertices when done. The problem there is that you could inadvertently move a boundary vertex of the area you are working on.

Ken