New bathtub diverter spout drains back into wall
How do you know water it leaking back into the wall. I can't quite see from your pictures, did you uses plumbers tape between the copper threads and the new brass piece (three times around, in the direction of the threads)? 

I've installed some of these spouts. They never have a set screw. Once the rubber seal is well inside the spout water shouldn't get back there, unless there is a leak between the copper and brass. 

The diverter isn't air tight. When water hits the diverter it gets pushed tight making a water tight seal, your breathe can't push it enough to make that seal.