What is the nature of the near field at a receiving antenna (excited by a far-away transmitter)?
Yes, antennas are reciprocal in every way.


  1) Does the impinging field (from a far-away transmitting antenna) cause the receiving antenna to develop its own near field, and what is its "shape" or "quality"?


It's the same as for any antenna.


  2)Continuing with this idea, is there a simulation anywhere such as this one made at MIT: 


Yes, it's the same simulation.


  3)If there is a derived near field at the receiver, does the receiving antenna then re-broadcast any energy?


Yes, but you can't distinguish it from the impinging signal. Its effects are seen as a modification of the field &mdash; a reflection or refraction.


  4)Extra credit: Does this touch upon Physics' "Arrow of Time" question?


No, not really.

Bascially, any conductive or dielectric object will interact with a propagating EM wave in a way that is defined by Maxwell's equations. You can think of the object as creating a local modification to the effective impedance of the space around it. The field causes charges to move, and moving charges cause propagating fields. It really doesn't matter "which came first".

The Feynman Lectures on Physics, particularly volume II, provide a lot of insight into this topic.