Well, actually, we really have no idea what caused the Big Bang, but it definitely wasn't the motion of subatomic particles! All we really know is that the universe is currently expanding, and our observations of the universe imply that this expansion started with some kind of incredibly strong explosion. But the important thing to keep in mind is that it's not the matter in the universe that's expanding today - it's space itself. That is, it's not the case that before the Big Bang, the universe was empty, and then all of a sudden something went BANG and created lots of matter. As far as we know, the Big Bang was the beginning of the universe itself - space, time, and matter. So if time itself began with the Big Bang, it really doesn't make sense to ask what happened before it!

Of course, the question is then to understand what caused the bang in the first place, and this is what we have very little understanding of. Some theories say that perhaps our universe is only one of many, and that a "collision" between our universe and another caused the Big Bang. Some theories say that perhaps it's not quite right that spacetime began at the Big Bang, but that there was a universe that existed before ours which collapsed and then "bounced" back out again to turn into our universe, and this bounce is what we see as the Big Bang. Unfortunately, it's really hard to test any of these theories, because it's really hard to make observations that far back in time!

Regarding the last part of your question, we also really don't have much understanding of where the laws of physics (like gravity) come from. Were they created along with the rest of the universe at the Big Bang? Or did they exist in some sense even without the Big Bang? I think most physicists would probably agree that the Big Bang didn't generate forces; it just created the universe and matter, and forces simply ended up being able to manifest themselves once there was matter around!

The beginning of our universe is one of the least-understood (and most interesting!) questions in physics today, so I can't really give you many definitive answers. But you're asking the right kinds of questions!

Before I answer this, I need to caution you: this is what Stephen Hawking THEORIZES may have happened. Nobody, including Hawking, actually KNOWS. Moreover, the theory in question that predicts this origin of the universe is extremely hard to test, so take this with a grain of salt.

The idea as I understand it, not being a quantum physicist, is that quantum fluctuations in vacuum or other state of non-existence create virtual subatomic particles that are partially real part of the time. This particular phenomenon has been observed with regard to electrons and what-not in high-energy physics experiments: if you collide a proton and an antiproton at high speeds in a particle accelerator, you will get all kinds of stuff that aren't parts of a proton or an antiproton, including things like electrons and positrons (and, of course, gamma rays, lots of them). If there is such a thing as quantum gravity, then vacuum should also contain virtual gravitational particles, and these particles may have been what drove the universe apart during the Big Bang.

However, herein lies a problem: based on our current best theories of space and time (general relativity as formulated by Einstein in the early 20th Century), time literally began during the Big Bang. There was no "before"; it's meaningless to ask what was before. The entire universe including space itself expanded out of something called a singularity, a spacelessness of zero volume, infinite density, infinite temperature, and infinite (anti-)gravitational strength that, in mathematics, is called a singularity (for clarity, you're familiar with the function
y = 1/x, which has a singularity for y when x = 0).

Singularities are impossible under quantum mechanics, so either general relativity or quantum mechanics (or both) is wrong, but since gravity is such a weak force it is impossible to test gravitational effects on quantum mechanics, and since quantum mechanics takes is limited to such tiny volumes, it's impossible for us to measure quantum effects on a scale big enough for gravity to matter. Consequently, we don't know which one(s) is wrong, or how it(they) are wrong, only that something here doesn't fit.

Hawking is a big proponent of quantum gravity, but unlike Einstein, who had actual data to back up his theories, Hawking doesn't. This is a limit of our technology, i.e. it's not Hawking's fault to not have data, but he doesn't. He's probably right that there is some quantum form of gravity, but exactly how it works and what role it had in causing the universe to come into being is not something we're going to know without some way of telling which of the many competing theories describing quantum gravity is correct.