What is driving the expansion of the Universe? Is it just the momentum from the Big Bang? If yes, how can it account for the acceleration of the expansion?
Why isn't the collective gravitational force slowing it down?
What part does dark energy play in the expansion?
As far as I understood it, no one really knows. It is observable that the universe is expanding with increasing speed, therefore there has to be a form of energy driving it. We can meassure (or at least try) the amount of energy and find $\Omega_\lambda = 69.11\%$ (see $\lambda$CDM-Model)

To your questions:

1: I wouldn't imagine it as Big Bang momentum, for it is also changing. After the Big Bang, there was (or at least had to be, to make our comological model work) a period of inflation, meaning rapid expansion, which then slowed down again. I can't think of an explanation which includes some kind of finite momentum.

2: On small scales (speaking of galaxies), where densities are high, expansion doesn't act. Gravity is working against it.

3: It is responsible for the expansion. How exactly is not understood. (Little personal comment form my side. If a physicist calls something 'dark', it mostly likely means it is not understood)