1. Damping introduces a second force. Friction. The definition of SHM is very strict. The restoring force on the object must be proportional to its displacement from the equilibrium position. Any other forces acting mean it's not SHM. Friction opposes the restoring force. It's a matter of definition.
If you damp an oscillation you increase its period. Always. If the damping is small the change in T isn't very much. This is why you get away with calling a mass on a spring in air SHM, because the damping is very light. (Try and get it to oscillate in oil!) Strictly speaking, it's only SHM if the motion is frictionless. (And the spring must obey Hooke's Law.) Just use your common sense. If you increase the damping of an oscillation, you will eventually prevent the oscillation altogether and the period will be infinitely large and it will never complete a cycle. (This is called heavy damping.)
I think you are assuming that anything that oscillates is SHM. This isn't the case. SHM is a very special case of oscillation. However, if you know something is SHM, or very nearly SHM, there's lots of lovely maths that can help you describe the motion and calculate useful things like period and frequency.
2.
The rubber barrier will oscillate as it's an elastic material. It has its own mass so the frequency at which it oscillates will be determined by that mass and the spring constant of the material. If you add lead balls you increase the mass of the barrier and therefore reduce its frequency of oscillation. The formula for the frequency of oscillation of a mass subjected to an elastic force says that if you increase the mass you reduce f and increase T.
If the barrier vibrates at a lower frequency it will resonate at a lower frequency. If this is the case it will resonate for lower frequency sound. (The low bass notes.) This means these low frequency bass sounds will tend to be absorbed by the barrier and not pass through it.