How does it work when you go from the binomial to the Gaussian, when the binomial is half-bounded from $[0,\infty]$ and discrete, whereas the Gaussian is a continuous distribution going from $[-\infty,\infty]$?

The key is that the binomial only starts to look like a Gaussian for $n$ very large. As $n$ increases, the binomial moves farther and farther away from the origin, so that its leftmost bound becomes less and less important (try it with the Mathematica code). Furthermore, the binomial distribution's discreteness gets less and less noticeable as more and more points get added with large $n$.