When Data mine is triggered, if the runner survives this ice, can (s)he/other still continue?
Core rules:


  After the Runner breaks all of the ice’s subroutines and/or any
  effects from unbroken subroutines resolve without ending the run, he
  has passed that piece of ice. He then continues the run by either
  approaching the next piece of ice protecting the server or proceeding
  to the Access phase if there is no more ice to approach.


Data Mine doesn't end the run, so the run continues.  In you example, the runner would go on to access Melange Mining Corp.

It trashes itself as part of its ability because trap ice cards are designed to be hard to break but only hit once.