What is the evidence that distant galaxies are moving away from us with speeds greater than $c$, due to space expansion? I came up with this query after @Rob Jeffries's answer to a previous  question of mine. 
So, is there any evidence that distant galaxies are moving away from us with speeds greater than $c$, due to the  expansion of space, or is it just an artifact of Hubble's equation, $v=H_0D$?
If indeed this is a fact, does it determine the shape/geometry of our universe?
 A: There is evidence that galaxies at red shifts greater than about $\sim1.5$ are indeed moving away from us faster than light. We can measure the red shifts, and observe their light. 
The General Relativity (GR) equation for redshift is not the same as it is in special relativity, which maxes at c for an infinite redhift. The relation of redshift to velocity is a linear approximation to both, and as you get to about z=0.5 and higher it starts diverging from the GR equation. V=HD which is Hubbles relation for velocity as function of distance and the Hubble constant, remains valid, even after v>c. Both GR and SR (special relativity) are approximately linear, v = cz, for velocity and redshift, at lower speeds. SR tops out at c, GR moves further up.   
This is all pretty well explained in books like Dodelson, and you can also see it explained and graphed very nicely (all three curves, GR, SR, and the linear relation) in the following Arxiv article. Figure 1 particularly graphs all 3 curves, but the article has the equations. See,
https://arxiv.org/pdf/astro-ph/0011070v2.pdf
This effect does not determine the shape or geometry, it is rather fully determined by them. Specifically, the Lamda Friedman Robertson Walker solution of Einstein's Field Equations (with a cosmological constant), with isotropy and homogeneity assumed, leads exactly to the observed values for z as function of distance, for cosmological distances. 
Edit: answer to @Yogi DMT's comment below.
No, the other way around. The crests are the same, except space stretches, and the distance between crests also. Say it emitted N cycles. It has to be the same number of complete cycles received, but the space covered by the same number of complete cycles is bigger, so the wavelength is larger. See a couple sites. 
This one gives a basic interpretation of the gravitational redshift, and including the cosmological redshift:  http://curious.astro.cornell.edu/physics/104-the-universe/cosmology-and-the-big-bang/expansion-of-the-universe/610-what-is-the-difference-between-the-doppler-redshift-and-the-gravitational-or-cosmological-redshift-advanced. It doesn't answer your question exactly but gives you a sense of the gravitational and cosmological redshift.
This next one gives more detail and detailed differences, a lot more specific, and also in the section on Expansion of Space derives the equation for the cosmological redshift in the standard cosmology model. It shows that 1+z  = $a_{now}$/$a_{then}$, where the a's are the scale factor of the universe, i.e. The ratio in the equation is the ratio of the universe sizes (or say the distance to a distant specific Galaxy) now and then. Now is when the light is received, then is when emitted, in the comoving time coordinate. This next one gives more detail and detailed differences, also in the section on Expansion of Space derives the equation for the cosmological redshift in the standard cosmology model. It shows that 1+z  = $a_{now}$/$a_{then}$, where the a's are the scale factor of the universe, i.e. the ratio in the equation is the ratio of the universe (or say the distance to a distant specific Galaxy) now and then. Now is when the light is received, then is when emitted, in the comoving time coordinate which is that used for the cosmological metric. It is at 
https://en.m.wikipedia.org/wiki/Redshift
You always have to be careful in these calculations to not shift coordinate systems in the same equation or concept. Easy to get confused, but the math is sort of easy. 
A: If the galaxy was traveling away faster than the speed of light, then we wouldn't be able to see it. (obviously) By very definition, it exist outside of the "observable universe". Not only can we not "observe" it with our eyes, but no information can reach us at all. (similar to the inside of a black hole). It cannot affect us in any way.
Asking wether things outside of the observable universe "exist" is somewhat of a "Zen riddle". It opens an very deep philosophical or meta-physical debate about the definition of the word "exist"
