Since we've barely worked out how exactly life appeared on our planet, trying to label one theory of the Universe's creation or structure as absolutely true is premature, at best.
I'm using slightly different nomenclature. Universe is something that exists in Space. Space is the emptiness, the Universe is everything else. I'm also standing by the "finite looped space, finite universe" viewpoint.
Also, the Big Band would be a perfectly distributed explosion. Since what exploded was a concentrated mass of pure energy, its composition was uniform. The parts closer to the center received less of an impulse than those at the edges, and the "impulse gradient", so to speak, scattered the energy/would-be matter in a near-uniformly filled sphere. From there, gravity and various perturbations thereof created the universe as we see it.
The "pulse of radiation" is a misnomer in case of ambient energy. For one, it would be a very very thick pulse. Its outer rim would consist of ultra-high-frequency waves, gradually receding in intensity and frequency closer to the center of origin. It'd also be a pulse of energy more than radiation, since it'd basically be launching into the void. Pressure, or whatever a force with similar effect would be called at that level, would push/pull the expanding energy into the surrounding space - this, unlike "stretching space", actually would increase the wavelengths.
And then, there's always the possibility that BR is being actively emitted. Gravity would have surely pulled some of matter/energy back into the center, creating a "universe core" composed of the physicists' worst nightmares (or wildest fantasies, whichever works). This could, theoretically, also be emitting this stuff, and the closed loop of the universe would mean that eventually it'd become uniform from all directions.