Can a runner make a run on an "empty" remote server?
No, remotes only exist if there is a card installed either inside of them (asset/upgrade/agenda) or in front of them (ICE). This differs from the central servers, which always exist even if there is nothing in or protecting them (e.g., archives).

This is discussed on page 4 of the official FAQ, under "Destruction of Servers" (emphasis mine):


  If there are no cards installed in or protecting a remote server, then
  the server immediately ceases to exist. If a server ceases to exist
  during a run, the run immediately ends. Unless the run has passed step
  4.4 of the timing structure of a run, it is not considered to be successful or unsuccessful.