Do "enters the battlefield" abilities trigger when a creature is copied with Phyrexian Metamorph?
You certainly do.  Per the Gatherer rulings:


  Any enters-the-battlefield abilities of the copied permanent will
  trigger when Phyrexian Metamorph enters the battlefield. Any "as
  [this] enters the battlefield" or "[this] enters the battlefield with"
  abilities of the chosen permanent will also work.


The section in the Comp rules seems pretty clear on this, as well:


  706.5. An object that enters the battlefield “as a copy” or “that’s a copy” of another object becomes a  copy as it enters the battlefield.
  It doesn’t enter the battlefield, and then become a copy of that 
  permanent. If the text that’s being copied includes any abilities that
  replace the enters-the-battlefield  event (such as “enters the
  battlefield with” or “as [this] enters the battlefield” abilities),
  those  abilities will take effect. Also, any enters-the-battlefield
  triggered abilities of the copy will have a  chance to trigger.