The universe is expanding at an ever increasing rate. Thus the following three-part question.
Are there any theories that suggest the expansion is a consequence of a pull from some unknown source rather than a push from the presumed dark energy?
Regardless of the cause, and assuming the expansion continues at an ever-increasing wheee, when will the most distant parts reach the speed of light?
And then what happens?
To take your second and third questions first: the Big Bang didn't happen at a point and the universe isn't a sphere expanding away from the Big Bang. Explaining exactly what we mean by the Big Bang is somewhat involved, but if you're interested I go through it in Did the Big Bang happen at a point?
That means it doesn't make sense to ask when the outermost bits of the universe will reach the speed of light. If we assume the universe is infinite then there have been parts of the universe moving away from us faster than the speed of light at all times starting the moment after the Big Bang.
So your questions (2) and (3) can't be answered because they are based on a misapprehension. However your question (1) remains interesting because there is a sense in which the accelerated expansion is indeed due to a pull.
Bearing in mind what I said about the Big Bang above, as far as we know the universe has no outside so there cannot be anything outside it to pull on it. However dark energy behaves as if it has a negative pressure, so at the risk of just playing with words if normal pressure pushes then we could argue that the negative pressure of the dark energy pulls the universe and that's what causes the expansion to accelerate.
However I must emphasise that this isn't a pull in the usual sense of the word. The negative pressure of dark energy just means that energy density of the dark energy doesn't decrease as the universe expands.
