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.