One more quick item that I should've mentioned above: My answers are somewhat misleading, because they could be taken to imply that you
require String Theory to answer your questions. They don't, necessarily. The problem is that your questions are really swirling around one question: What caused the Big Bang? In other words, what caused the initial expansion. If you simply assume the initial expansion, as physicists necessarily do, then the explanations for your questions simply follow (from General Relativity). However, in order to explain the initial expansion itself, upon which the other questions are ultimately contingent, then you require something more, and that's where String Theory fits in.
Having assumed the initial expansion, however, the answers are what I already gave you: gravity (plus cooling, to put it simply); dark energy (or whatever you want to call it); and physical laws weren't different (the universe didn't 'travel' faster than light; rather, timespace was stretched and light stretched with it).
"Having assumed the initial expansion, however, the answers are what I already gave you: gravity (plus cooling, to put it simply); dark energy (or whatever you want to call it);" 
No, gravity is what prevented the Big Bang from happening any earlier...it was the force that originally bound all matter close together (remember that space itself exerts no force) per the inflation theory.
Nor did cooling happen. Cooling is a colloquialism. In reality, to cool an object you must move heat away from said object. This is why heat radiates away from a hot stove, i.e. the heat is moving away from said stove. Turn the stove off and eventually all excess heat will radiate away into the environment.
But the universe has no such place to move heat outside or away from it. Energy is neither lost nor made, after all.
So the universe didn't "cool" overall...all of the same heat was and is still in it (unless you can identify where it went). Note also that empty space can't hold heat (though heat could transit across it in some fashion).
And dark energy doesn't exist. We've never seen it in the wild or in the lab. It's a mathematical figment...a crutch...a tool of a mathematical model.