The Universe started with nearly the same initial conditions in all directions and distances. As a result, cosmic time was nearly synchronized everywhere we look. This means that the Big-Bang instant in our past, when the density of matter and radiation diverged, occurred at the same time everywhere we look. As a result, light traversed the same distance towards us since that singular beginning from all directions. As we look out, our cosmic horizon is a spherical boundary, defined by the distance that light traversed over the past 13.8 billion years. That the initial conditions were similar out to much greater distances is a moot point because we cannot access these regions.
The Universe was transparent to gravitational waves since the earliest Planck time of its expansion history. But the thermal background of gravitons is extremely difficult to detect, as explained in my paper with Sunny Vagnozzi here.
Yep.
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