Depends on the franchise, but mostly, I'd say yes. In certain franchises zombies deteriorate rather quickly via rotting. A penis is about the most easily losable extremity on it's body.

And to the most pressing question, you can certainly slice it off with a showel and if you somehow manage to get a clean swing you could deliver enough force to separate it from the zombie's body via a blunt force trauma. That would however be extremely difficult as ther is barely any position/angle that would allow you to do so outside of extreme luck.