Faster-than-light gravitational waves and faster-than-light expansion in the inflation I have no introduction to the inflationary epoch. I know, however, that during this time space-time expanded with a speed faster than the speed of light. If gravitational waves are perturbations of spacetime how is it possible that their speed is limited by $c$ and at the same time there is no restriction, in terms of speed, on what spacetime itself is allowed to do?
 A: 
during this time space-time expanded with a speed faster than the speed of light

This statement actually is the problem here. The expansion of the universe is not measured in units of speed, so it cannot really be compared to c in the first place. Saying that it is faster than the speed of light is “comparing apples and oranges”.
The expansion of the universe is currently about 70 (km/s)/Mpc. It was much larger in the inflationary epoch, but would still have the same units. So even then it does not make sense to compare the inflation rate to the speed of light. There is always a distance where the expansion between two points separated by that distance is less than c.
In contrast, the speed of a gravitational wave is an actual speed. Even on a local scale a gravitational wave travels at c. This is important because in GR only local speeds are physically meaningful. Speeds of things that are not colocated are not even well defined in a curved spacetime.
The speed of a gravitational wave is local, and therefore meaningful, and is c. The expansion of the universe is not a speed and cannot be converted into a local speed other than 0, so it is not meaningful and therefore cannot meaningfully be compared to c.
A: There is no such thing as a limiting speed c for gravitational waves!
With this question, you have hit on an important topic that is usually badly discussed in lectures, articles and books.
There is no general law that says that the velocity of gravitational waves and gravitational perturbations is necessarily c. Nor is there any law that says that they cannot exceed the limit c. Yet they are almost always quoted as if they were proven theorems!
The misinterpretation originates from the fact that Einstein derived the speed c for a special type of gravitational waves - by the so-called linearized Einstein equations. So it is a special approximation when the spacetime over which the waves propagate can be taken as Minkowski spacetime and the intensity of the waves is weak. Clearly this is an important but very special type of all gravitational waves and effects.
It has been 100 years since then, and no one has been able to produce a similar derivation for all gravitational wave and effect propagation cases. But many researcher have tried many times... For some other special spacetimes there are similar approximate results.
But the question of the velocity of propagation of weak and strong gravitational waves and perturbations of curved spacetimes remains open, despite the false myth.
In fact, in advanced research that attempts to describe gravitational effects propagating in curved spacetimes ( coupled to material fields), it is usually concluded that the waves propagate at a speed other than c! They do not have a uniform velocity, but have a wide range of velocities depending on the couplings, the parameters and the way the approximation is made. They can propagate at speeds less than c, but also at speeds greater than c. This is all the best theoretical models can tell us for now. See for example this source, I highlight one statement in the abstract:
"We show that for the choice of interaction signs implied by S-matrix and spectral density positivity bounds suggested by analyticity and causality, the speed of gravitational waves is in general superluminal at
low-energies on NEC preserving backgrounds, meaning gravitational waves travel faster than allowed by the metric to which photons and Standard Model fields are minimally coupled. "- The Speed of Gravity
