What would be the effect on gravity if space expanded at > $c$? If space were to expand at > $c$ (as in inflation) would that mean gravity would no longer have any effect on the curvature of space, since gravity can only propagate at $c$?
 A: Your question is ill-formed for several reasons.
First, inflation is driven by gravity: all stress-energy gravitates, and if this is a field with certain properties, you get inflation.
Second, "gravity propagates at c" refers to a perturbation on some background geometry, relative to that geometry. For example, if you wiggle some mass or whatever, the gravitational influence that produces won't be superluminal. But that does not mean that two sufficiently distant particles can't increase their distance faster than c due to expansion of space.
Third, inflation is not characterized by superluminal expansion per se. It refers to an exponential expansion in the early universe. It is completely possible for the proper distance between two galaxies to be superluminal without space expanding exponentially (although given dark energy, our universe is probably transitioning to an inflation-like exponential expansion).
A: 
would that mean gravity would no longer have any effect on the
  curvature of space

Gravity is the curvature of spacetime.
Let me emphasize this:  it isn't that gravity affects the curvature of spacetime; it is that gravity is the curvature of spacetime.
A: I think the question is ill-stated for a simple dimensional
reason. Expansion rate is measured by the Hubble constant which is the
inverse of a time (speed/distance). It cannot be compared to an
actual speed such as c, which is distance/time.
It is as meaningful as saying that the speed of inflation at Wall Street
is 123 mph. The use of the word "speed" in place of "rate" is common, but ill-advised.
