Did the Big Bang singularity happen in everywhere in the space that now exists today?

Yes.

If you were to run the big bang back into time, for a closed universe, you would see that the volumn of 3-d space at each time step decreases during the 'collapse' but since the number of fundamental particles and radiation quanta remain the same, because they were produced at the instant of the big bang, the DENSITY of matter and energy in 3-d space increases.

Mathematically, in such a universe, you can calculate exactly what the volume of 3-D space is at each epoch. As you near the big bang itself, the tiny neighborhood surrounding every point in 3-d space becomes smaller and its density of energy rises. At an infinitessimal instant after the big bang in the classical theory of big bang cosmology, the density of these vanishingly small neighborhoods of 3-d space increase to infinity, but do so in a still-finite number of points that remain to define the 3-d closed surface of the remaining space.

In an infinite universe...which has always been infinite in its 3-d spatial volume even at the big bang...as you run this collapse back to moments after the big bang, you are still left with an infinite volume to 3-d space, but the density of matter and energy in the neighborhood of each point, becomes infinite. So, unlike the nearly point-like initial 3-d state of the closed universe with its nearly point-like initial singularity, for an infinite universe you end up with an infinite surface which has an infinite density of energy and mass at each point.

This 'classical' description of the initial singularity breaks down when you add quantum mechanics to the picture. The minimum size becomes the Planck length of 10^-33 centimeters, and the density is no higher than 10^95 grams/cc. So, in both open and closed cosmologies, you reach a limiting density of 10^95 gm/cc everywhere. Also, various kinds of phase transitions may come into play to produce inflation once the universe reaches a size of 10^-25 cm.

So, in both cases, the answer to the question is rather hard to come by because in a mathematical sense, all the points in today's space existed in the original space at the time of the big bang. Only their separations have changed.

In deeper detail, space is just another name for the gravitational field of the universe which is, fundamentally, a quivering soup of virtual particles...gravitons...which come and go. How these gravitons are supposed to relate to the idea of 'points in space' I don't think is very well understood until we can perfect a quantum theory of gravity.