No "whoops" about it.
First, your calculations presume that the universe itself (as distinct from its contents) is a) flat and b) not expanding. Neither presumption is true. You're looking at things as if they were moving along a straight, flat, fixed highway, traveling east and then sending signals back by light-speed carrier pigeon.
Instead, you have to understand that the universe is like a balloon being blown up bigger and bigger, and the objects in it (including us) are ants on its surface.
Imagine that we're standing on the balloon's "north pole". Where are the objects which are the farthest possible distance from us? They're near the south pole (and incidentally all near each other).
Now deflate the balloon so that we can see how things were shortly after the Big Bang (when the universe was much smaller "around"). The most-distant objects weren't all that far from us then. But we aren't yet able to see them as they are at that instant, because light travels by little carrier pigeons (photons) that crawl along the surface of the balloon at the speed of light.
If the balloon (universe) had stayed small, we'd have been able to see them pretty shortly thereafter, but there was a catch. As the carrier pigeons were travelling towards us, the balloon was being inflated quite rapidly. As the carrier pigeons trudge along, the distance between them and us keeps getting stretched larger.
What would have been a few light-year journey had the balloon stayed the original size ends up being a journey of several billion years as the poor carrier pigeons keep trekking along at light-speed as the balloon keeps getting bigger and bigger and bigger as they travel. Eventually, though, they make it to us, and we finally get to see what the most-distant objects looked like many billions of years ago (back when the universe/balloon was young/smallest).
The balloon has been expanding for about 15 billion years, but it's still possible to see light from 10 billion (or more) years ago coming in from an direction, because that light has been fighting an uphill battle against the expansion of space all that time, even though the objects the light is coming from weren't all that far away from us (or each other, or anything else) at the time the light was emitted.
This is also how we can see light traces from the earliest moments of the Big Bang itself by looking far enough away in *any* direction (the "background radiation") -- we're seeing light arriving from the "south pole" (farthest/oldest possible origin) from back when the north and south poles were pretty much right on top of each other. This is like if you set off a huge explosion at the Earth's south pole, the sound of it would arrive at the north pole simultaneously from all directions.
I hope this helps clear things up some.
LOL, oh sure, real clear, right up until that moment one of your damn pigeons crapped on my glasses!
Your balloon-pigeon analogy, like all analogies are only an image of reality, and at some point breaks down into meaninglessness. Take for instance the presumption(?) that all the matter expanded at the same rate from the moment of the BB (i.e., all matter was on the surface of the balloon). To use your balloon analogy, I would expect the balloon to have and infinite number of concentric inner layers, all having some amount of the matter on each of those balloon surfaces.
--Boot Hill