Does our Galaxy move at the speed of light? I am not a physicist, but I have a few  questions that bother me for a while:


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*As we do not know if the Universe is infinite or not I assume we do not know our whereabouts in it;

*We know,though, there is an observable universe that expands at different rates;

*If we mark our galaxy with A, is it possible for galaxy B somehow “behind” us in the Big Bang funnel shaped graphic to see A on the edge of its visible universe? 

*Is A’s light red shifting for B? 

*Do we move away from B at the speed of light?

*If Universe expends how come our solar system does not?
 A: The Hubble constant is the measure of how fast the Universe is expanding today and its value has been measured to be 70 km/s per Megaparsec (a parsec is just a unit of distance equal to about 3.26 light-years, and a Megaparsec is a million parsecs). This means that on average, for every Megaparsec two galaxies are separated by, they are moving away from each other by 70 km/s. Therefore, to be moving away from each other at the speed of light, two galaxies would need to be separated by a distance of about 4,300 million parsecs. This is smaller than the radius of the observable Universe, therefore not only are there galaxies in the Universe that are moving away from us faster than light, but we can still see them!
A: Question: Does our Galaxy move at the speed of light?
I would like to add to what Abhinav already has explained that the notion that galaxies are moving away from each other is a bit tricky. You could say as well that galaxies don't move but the space between them expands. The background here is that these interpretations are coordinate dependent. What expansion really means is that the distances between comoving object are increasing. Then all observers will agree. 
Question:  -If Universe expends how come our solar system does not ? 
The short answer is that gravitationally bound systems don't participate in the accelerated expansion of the universe. Only very large sytems called superclusters of Galaxies "feel" the accelerated expansion. If you like to know more  search Late-time integrated Sachs–Wolfe effect.
Why don't gravitationally bound systems expand? The FRW cosmology which describes the dynamics of the evolution of the universe is based on the perfect fluid model so that the stress-energy-tensor is non-zero locally. So according to this model an arbitrary observer would see the universe homogenous and isotropic independent of the scale.
Our universe isn't homogeneous on small scales but we can apply the FRW model on large enough scales where homogeneity can be assumed . On small scales - say the solar system - we can apply the Schwarzschild solution instead, as we have a central mass and vacuum (means the stress-energy-tensor is zero). As the energy density of the dark energy (the cosmological constant resp.) is non-zero locally, it is - if I remember that correctly - in principle possible to calculate the tiny stretching of the solar system due to the dark energy. This can't be understood as expansion though. The action of the dark energy on the solar system is a tiny not measurable increase of its size which is constant over time.  
Question: "Is A’s light red shifting for B?" 
Yes, comoving objects (galaxies) see themselves red-shifted due the increasing distances between them.    
