I turned off snap to grid and the square moved correctly, yay. Thanks, the not snapping to the grid problem was the too small subdivision size of the grid which was set at 1 mm. I guess the 3px selection tolerance must be bigger or the snap to point tolerance of 5 px is bigger than the 1mm. If the grid lines are set to 10mm with 2 subdivions then snap to grid works again.
Or in inch terms, I had 0.4 inch gridlines and 10 subdivions of that or 1/32 inch. Now it is set at 0.4 inch grid lines and 2 subdivions and snap it does.
Sorry to bore, but I write it out again if In case I forget;
Turn align to pixel grid off otherwise snap to grid does not work at all, either at the start of a new document or inside the transform panel. Once this is done;
Smart guides only work if snap to grid is turned off, or you cannot have snap to grid + smart guides together (why the hell not?)
Keyboard increments set to large say 10mm or 0.5" help alignment a lot, but if snap to grid is turned on they are off, like one increment + one more subdivision with each keyboard stroke, so instead of 10mm it is 11mm (when I had 10mm gridlines and 10 subdivisions) or currently 10mm gridline + 1 subdivision= 15mm (I have 10mm gridlines and 2 subdisions or 5mm). If snap to grid is turned off they are exact.
Snap to grid does not work despite the above if the grid is too small, gridlines at 10mm and only 2 subdivions anything smaller and snap to grid does not work.