Interesting thoughts. The reason that the Universe has no edge and no centre but has finite size is that it is (on the largest scale) a 4 dimensional hypersphere. (4 spatial dimensions)
The best way to visualize this is imagine you are a bacterium on the surface of a planet-size smooth sphere. As far as you are concerned the surface is flat and you have the ability to only go right and left and forward and backwards. If you were to travel sraight forward you would come back to where you started and during your travels you would never see an edge or a center, but you would have gone a finite distance. So the Universe appears to you as 2 dimensional (flat) but is curved in the 3rd dimension (the sphere). Our Universe is the 3D version of this. It appears 3 dimensional (flat, as has been recently proved) but is curved in a 4th (spatial, not time). Thus there is no center and no edge.
You also ask how come no matter where you are in the Universe, there is always redshift on all directions? Lets go back to our bacterium on the sphere. Imagine the sphere is slowly expanding. To the bacterium, no matter where it is on the sphere, in every direction is always moving away from it. In our Universe every point is on the "surface" of the hypersphere and thus no matter where you are, everything appears to be moving away.
As far as the Big bang is concerned, it happened as the 3 Kelvin background radiation is proof of this. In the steady state, there would be no background radiation. At the big bang time=0, and the 3 normal dimensions where also 0. The 4th spatial dimension was infinite as the curvature of the Universe was infinite.
Black holes are miniature versions of the Universe put in reverse. At the centre is a singularity where density is infinite and time=0. Also the dimensions of space are also 0.
This is my interpretation of the Universe. It may or may not be right. But then again, no one really knows how it works!