Good questions, lean bean. I'm not a general relativity expert but I'll answer to the best of my ability.

The big bang is generally called a singularity because our equations treat it as an infinitely dense state--that's an infinity and since infinity isn't a number we can't really get numbers out of our equations, so we say they're singular or that this is a singularity. I believe this happens at the big bang, as plugging in any finite numbers will likely get you a number out. The bigger question is what happens when you get "close enough" to the instant of the big bang. Is there some extremely large but finite density at which general relativity is not longer an accurate theory? I think most physicists believe so and that's why they're hunting for a theory of quantum gravity, which would cover the very tiny length scales that matter at those densities.

2) Yes, as I understand it the observable universe would be smaller in the past. This is because only so much time has passed since the big bang, so light can only have traveled so far to reach us. The less time between us and the big bang, the smaller region of the universe we can see. (Expansion plays a role as well, so we actually see stars that have since moved further away due to expansion). The closer you get to the big bang, the smaller the observable universe would be. At 1 picosecond after the big bang, assuming we could be there to "see" objects, we'd see things 1 light-picosecond plus whatever expansion had happened distance from us. (Obviously we can't necessarily imagine "seeing" things in this early universe since it was so hot and dense, so what we're really calculating is the maximum distance at which things can interact with each other since all interaction travels at the speed of light or less).

One of the toughest things for me to grasp about this was that if the universe is currently infinite in size, but has finite average density, the big bang could have been infinite in size with infinite density. It would still be a singularity because infinite density makes the equations break down. Maybe Pete can correct me on this, but I was under the impression that if the universe was finite in size (even at the big bang) it would still have to be infinite in size, and if it was infinite in size at the moment of the big bang, it would still be infinite in size.