The Big Bang isn't a "thing", it was an event. Similar to inflation, which was another event soon after the BB. The Universe was once very very dense and expanded from that state into a less dense state. Our current models only take us back to a certain point in time after the Big Bang. Before this point in time our models say the universe was so dense that the model starts making infinities, thus it "breaks down" at that point and predicts nothing. The singularity is nothing more than our model breaking down. (Or so most cosmologists believe I am told)
So your question about the size of the BB and if it popped into existence should be reworded as the size of the Universe and such. And our only answers are: "We don't know"