Question:  Does an Aether exist, and what are its properties?

The standard model of physics neither requires nor prohibits the existence of an ether.  A perfect vacuum is known to contain a "quantum foam", where virtual particles pop in and out of existence, which is itself an ether or a medium that permeates otherwise empty space.  Relativity theory does not describe an ether, but Albert Einstein states that space is filled with an ether. [1]  

2004 Physics Nobel prize winner, Frank Wilczek penned the term "the Grid" to refer to a modern understanding of ether in his 2008 book "The Lightness of Being, Mass, Ether and the Unification of Forces". [2]

Conclusion:  The vacuum of empty space is not actually empty, it is teaming with virtual particle that pop in and out of existence and may have many other properties that are not yet well understood.  (See related speculative theory Ether gravity).  

[1]  Albert Einstein said that space is "endowed with physical quantities", but that "this ether may not be thought of as endowed with the quality characteristic of ponderable media [...] The idea of motion may not be applied to it"—from Einstein, Albert: "Ether and the Theory of Relativity" (1920), republished in Sidelights on Relativity (Dover, NY, 1922)

[2] FRANK WILCZEK, THE LIGHTNESS OF BEING: Mass, Ether, and the Unification of Forces (Basic Books; September 2, 2008)