When cell dragging is recorded as a macro, the resulting VB code uses select, cut, copy and paste, depening on whether ctrl is used to duplicate, and whether alt is used to move to
another sheet. Also, dragging a cell clears the paste buffer. Both of these seem to indicate
that cell dragging really uses cut-copy-paste behind-the-scenes, so it's unlikely you can move
the contents of a cell without using cut and paste, even if you use the mouse.
I'm going to try and answer this question in parts.
For your cell formatting: What gets moved is the formatting on the CELL, not what it looks like. Let's take cells A1 and B1. They're next to each other. There's a border between the two. I'm going to move cell B1.
If the border is because B1 has a left border, it'll move.
If the border is because A1 has a right border, it won't move.
I never use cut, because it ISN'T copy-paste-delete. Cut is treated as a "special" action, and as a result, it ignores filters. It will also move cell references with it, and overwrite cell references where it lands. If you try similar experiments with paste special data into filtered ranges, or if your drag-down is copy then paste special, you might see similar behavior.
I've had quite a few problems at work from people cutting and pasting data. My mantra is "Never cut and paste - you won't get the results you're hoping for"
Best Answer
Select the row that you want to cut. Press Ctrl + X. Marching ants (marquee) appear around the range.
Select the cell to which want to move the data.
Press Ctrl + Shift + + .
Note that this may only work if the source and destination ranges are along the same column.