The Big Bang lies in the past light-cone of every event in spacetime, so in one sense the Big Bang is the ultimate cause of everything. But this somewhat begs the question because we do not understand exactly what the Big Bang was, and we probably won’t until we solve the problem of quantum gravity. What we do understand is the state of the universe a very short time after the Big Bang and the laws that have governed the subsequent evolution of the universe.
As other answers have pointed out, this does not imply that the universe is deterministic. The answer to that depends on your chosen interpretation of quantum mechanics. In your domino analogy this is like a domino run which starts in one place but has many branches. We know where the domino run starts, but we don’t know whether the choice of direction at each branch point is pre-determined. Indeed, in some interpretations of quantum mechanics there is no “choice” required, because the dominoes fall in all branches at once.