The effects of gravity spread at the speed of light. If the sun suddenly blinked out of existence, it would take eight minutes to both see it disappear from Earth and notice a change in our planet's motion. In other words, the planet would continue to orbit that empty space in the middle because the sun's gravity, much like it's light, would still be extending out to us.
The universe is expanding in every part of space all at once. Some places are so far apart that the collective expansion between them is growing at greater distances in a given amount of time than light can travel. And the same can be said for the effects of gravity, but the motion of objects caused by gravity at great distances is far, far slower than the speed of light.
Our galaxy is moving in a direction that is caused partly by something called "The Great Attractor." But even though it's causing us to move, we will never reach it, and it's influence on us will gradually weaken. And it's not even that far away compared to most of what we know is out there.
That IS the heat death of the universe. It's not heat death as in "a wave of heat destroys everything" or anything like that, it's heat death as in "there is no more heat(aka condensed energy), everything is equally cold(aka lack of condensed energy) everywhere. Heat itself is gone, has died".
Could it simply be that all the light from the edge of the universe is finally perceptible to us? Have we literally started seeing the beginning of time itself?
Or is it more likely that I have no idea and am just posing questions so that someone can educate me on the matter?
No, it's definitely something different. "The beginning of time itself" can't be seen in visible light due to the heat in the early universe - the CMBR is the only thing we see from that period.