It doesn't make a difference if the filter is new or old during the "hose disconnect flush".
But, if you pull the pan to remove the filter after the flush, you'll be wasting the new fluid in the pan. If you pull the pan before the flush, you'll dump the old ATF which is a better way to start a 'hose disconnect flush'.
If the new/old filters are the same, then NOTHING should be left behind unless the old filter failed(which is unlikely).
Filter types are application specific. Not all are screens. Some trannies have screw on filters. Others use a "pita pocket" type filter cause it looks just like one and the pocket is where all the crud collects. And, some have a flat felt oil filter type media.