Scott is correct. There is no way to have no tool selected. This is the same in GIMP and other raster image editors like Photoshop. Also, Select > None simply deselects a selection. It has nothing to do with the tools.
I think the main problem I see with your screen shot is that you have a "Floating Selection" active. GIMP is different from Photoshop in this respect, and this behaviour can confuse people who've only used Photoshop.
So, now all you need to do now is to decide what to do with the Floating Selection.
Either: attach it to the existing layer by hitting the Anchor icon in the layers panel. But this will attach it back to the image layer, and that probably isn't what you want.
Or: hit the New Layer icon in the layers panel, to anchor the Floating Selection to a layer of its own. Then you can use the Move Tool, to move the image where you like.