Yes, it's hard to initially grasp this, but the particular issues you're concerned about actually work out fine in the current picture. Let's look at the key points.
In the BB picture, the universe expands out not from one "given point" but from any given point. Stand anywhere. You'll see the stuff near you moving away, the farther the faster. Think of how things look from somebody else's point of view. They see the same thing.
One illustration often used is a raisin muffin expanding as it cooks. From any raisin's point of view, the other raisins are moving away. There's no particular place that gets especially empty, so there's nothing balloon-like about it. The red light shift is approximately proportional to distance, so it covers a huge range.
As for the current structure of the galaxies, on a fairly large scale it is indeed more spongy and irregular than one would get from well-stirred muffin dough. However, on a very large scale it looks quite uniform.
There is indeed a close tie-in between the BB picture and the current distribution of matter. At one stage, there were just small ripples in the density. These are still visible as small ripples in the cosmic microwave background (CMB) coming in from different directions. Over time, those ripples would tend to grow because regions with a little extra mass pull more mass in via gravity. That process can be simulated on a computer using ordinary gravitational dynamics. It turns out that the slight ripples in the CMB imply that now the matter distribution should have unevenness very close to what we see in the galaxies. So it really does all tie together.
The major remaining uncertainties concern what happened at even earlier stages. There the evidence starts getting thinner. For example, although the ripples in the CMB are close to what's expected from quantum fluctuations, we don't know for sure what was going on when those fluctuations set in. The main picture has been "inflation", a period of rapid exponential expansion, expected from General Relativity for certain types of transient physical states. (We're currently in a period of much slower inflation.) Problems with that picture have led to alternatives, including collisions between entire 3D spaces in some higher dimension. Weird as all that may sound, the Planck satellite is currently measuring details of the CMB ripples, in an attempt to sort out those possibilities.
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