To the original title/question: loading a rope to its breaking strength constitutes the one cycle to failure data point. Running 10 or 20x less than that load turns the rope into "huge number of cycles before any possibility of failure". So depending on the rope's history, you are dipping into its unknown durability reservoir by doing the test. Per an earlier post, a sort of statistical sample can occur by break testing a sample off perhaps one end of the rope. So, short answer, yes, strength testing the rope does cause damage by using up some of the cycles to failure, worsened by how hard you test the rope.
take care