Is there a universal rest frame of reference? I am still struggling with C being a constant and what that implies.
So can an experiment be done to find the resting state for the universe?
Take a device with an observer and a light source and two mirrors, one 10 meters in front and the other 10 meters behind the light source.  Now move this through space at some speed.  Can’t you determine your speed by using the red/blue shift from the light reflected by the mirrors?  Will this allow you to find the resting reference for the entire universe?
 A: You've missed something important about relativity.
The rule that all (inertial) observers measure the speed of light in vacuum to be the same is really just a special case of the big rule: all inertial frames have the same physics.
That is explicitly a claim that there is no experiment that can distinguish one free-falling frame from another without reference to some external frame of reference.
So no: there is no universally special frame of reference.
A: If the source and detector of light are moving with you, there will be no Doppler shift.
A: This experiment will not produce the result you expect, because redshift caused by the Doppler effect depends on the speed of the source relative to the observer. In your experiment this speed is zero.
See also the Principle of Relativity and Einstein's "On the Electrodynamics of moving bodies".
A: There is not a universal rest frame.
There is, however, a galactic rest frame. Because you can look up at the stars, falsely assume that they do not change, and count your rotations that way. However, that method is only as reliable as the premise that the stars don't move, which they do slightly. The Hafele-Keating experiment used a variant of this, utilizing the Earth's course through the solar system. The airplane headed west took a shorter path through the solar system than the one to the East, thus providing a 4d solution to their trajectories, and demonstrating the actuality of time dilation. The same tactics could be applied to measure the Earth's trajectory through the Milky Way, or the whole she-bang. But these only measure Earth's trajectory in respect to other physical objects, none of which are a universal reference frame.
As of now, 'the universe' and 'all the galaxies that are and ever were', are considered the same celestial object. But that's not what is meant by the term 'universal.'
A: This is an interesting question, one which should be posed very carefully. If I were to ask: "Assuming Lorentz invariance (relativity), can one measure a preferred reference frame for the universe?" the answer would be no, as explained by some of the others here. However, Lorentz invariance is a fragile symmetry, which is broken in our universe due to the presence of the microwave background radiation. So if I just asked: "Can one measure a preferred reference frame for our universe?" the answer is yes. Just measure the microwave background radiation in all directions. The preferred reference frame would be the one in which all directions give the same temperature.
A: First let there be an observer and a light source, at rest relative to each other. 
Now we accelerate the light source away from the observer. The observer says: The light from the light source red shifted.
Then we accelerate the observer towards the light source. The observer says: The light from the light source blue shifted.
If the two accelerations are equal, then finally the observer says: Now the situation is the same as at the beginning, except that the light source is a little bit further away, and if at the beginning the light source and I were standing still, then now we must be moving. 
A: Yes.  All the physics books say no, but the answer is Yes.  It's the reference frame from which light propagates at the same speed in all directions.  To clarify; light is a series of waves that can be considered to be pulses.  Each pulse propagates from it's exact location.  Motion of the light source does not affect the speed of any one pulse.  But the motion of an observer does.  The reference frame that coincides with the actual instantaneous pulse is the "Universal" reference frame.   
