Okay, the title is slightly misleading ..

You can not move layers without touching them, but you can move them from the layers palette without clicking on them using the following keyboard shortcuts:

To move a layer down to the layers palette, press: Command-left bracket (PC: Ctrl-left bracket)
To move a layer to the palette of layers, press: Command-right bracket (PC: Ctrl-right bracket)

Now, if you want to move a layer within the image without using your mouse, select the Move tool and use the arrow keys on your keyboard.

You are probably already familiar with the History of Photoshop, which allows you to undo up to 20 steps. Unfortunately, when you quit Photoshop nothing is saved in memory and this no longer works. However, there is a way to undo changes to color and tone after days, weeks or even months.

Here is how: next time you apply a change in tone of any kind (e.g. through options Levels, Brightness/Contrast, Curves, Hue/Saturation, Color Balance or other), do not select them from the menu. Instead, scroll to the bottom of the Layers palette and click on New Fill or Adjustment Layer. This is the half black and half white circle. You will see a pop-up menu from which you can choose what tonal change (or filling) you want to apply. You’ll see a new special layer in Layers palette with the name change of tone (e.g. Color Balance). When you save the document and open it again, the Color Balance layer will still exist. To edit the settings of the Color balance, double-click on it. To undo the action, drag the Color Balance layer to waste.