The second part of your question suggests a misunderstanding. Yes, gravitational waves are like light, traveling at the vacuum speed of light. But no, it does not mean that we can only catch gravitational waves "created much earlier than the inflation". On the contrary, you tell me how old the gravitational wave is that you want to catch and I'll show you the spots in our universe from which gravitational waves traveled just that amount of time to arrive here, just now. The same applies to light... in particular, the cosmic microwave background remains visible, as you see the glow of ever more distant regions of primordial gas while the universe ages.

But this has little to do with gravitational waves and inflation. In that case, what we expect to see (and we thought we saw, with the BICEP2 data last year) was the imprint of quadrupole gravitational radiation on the polarization of the cosmic microwave background. So it is not the gravitational waves themselves that we would be observing, but changes in the microwave radiation as it travels through a cosmos that is "filled" with the gravitational wave remnants from the inflationary era.

During inflation there was a violent expansion of 3-D space. Such a fast expansion may generate gravitational waves. However, the recent observation of gravitational waves which was observed by the LIGO experiment (2016) is a result of a binary black-hole merger 1.3 billion light years away. Therefore it is not the primordial type of gravitational wave which is being detected by LIGO.

In 2015, BICEP2 experiment detected polarization patterns  in cosmic microwave background photons. This was thought to be a signature of primordial gravitational waves created during the inflationary era. However, in reality this was an effect caused by interstellar dust background.  

However, it is hoped that in future  LIGO will be able to detect gravitational waves of primordial origin, which will shed light on the Physics of inflation.

An old but useful reference is:

CALDWELL and KAMIONKOWSKI, Scientific American 2001. Title of this article is "Echoes from the big bang".

This is a very standard material, available in a number of text books.

Inflationary cosmological model was proposed by Particle Physicist Alan Guth and others in early 1980s. Essentially, at high energies, as particles behave as quantum fields, contribution of a quantum field to the energy density and pressure need not be zero. In the Planck era the lowest energy state of this quantum field might have corresponded to a false vacuum, generating a cosmological constant which is  10100 times larger than what it is today, and resulting in a super-large repulsive force. Such a repulsive force stretches 3-D space which doubles in volume in every 10-43 seconds. After 85 such doubling (in volume) temperature drops from 1032 K to near absolute zero. 

Ref:

Hi Fabrice,

with ongoing cosmic expansion (accelerating at that) by the standard cosmology model, the visibility and reachable limits would be surprisingly close.

see http://www.astro.princeton.edu/universe/ms.pdf

The Pearlman SPIRAL cosmological redshift hypothesis and cosmology model should help resolve this.

In it the universe starts by a singularity (so a big bang hybrid), includes a cosmic expansion inflation event, into a stable universe ( no subsequent ongoing cosmic expansion OCE) that is a sphere, that approximates the size of the visible universe, 

with no OCE, matter traveling at light speed toward us from the most distant places could reach us in the number of years = LY distance to go.

The following thread may be useful in this regard.

https://www.researchgate.net/post/How_can_Gravitational_Waves_Detection_is_going_to_benefit_our_society?_tpcectx=profile_questions