Is it possible to copy and paste a selection?

Hi Keith.  Yes, so long as it is raster content like an image layer. 

To copt the entire layer you have to select it first (Ctrl A)

Ctrl C to copy.

If you use Ctrl V to paste, it will create a new layer.

Just one thing.  I see you have used the Android tab for your post.  Can you confirm what version of Photoshop you are using, and on what operating system?