You have to remember that cause often precedes effect in a time war, so it may not have a start point. It could have been started by an event caused by it's own effect, or an effect evented by it's own cause, or even a cause effected by it's own event. This makes Time Wars a near impossible event to study in terms of causes and links, and you have to study it entirely in terms of thing that may have happened and ignore any preconceived notions of the linearity of time.

One of the earliest thing that may or may not have happened was the failed/successful assassination of the entire dalek species, at the beginning of their creation. Apart from that, all the techniques to study such temporal events was lost in the destruction of Gallifrey.

As for what popped out from the other side, some sketchy sources say that the Doctor sealed off the war and destroyed everything inside it using a powerful weapon known as the moment, but this is only gathered from second hand testimony.

Daleks fricken daleks and their obsession with extermination that is all their need for fire fueld millenias of war.

The doctor survived by running that is all he could do. The daleks survived because all of the other time lords mysteriously dissapeard and they didnt continue with their conquest because they were searching time trying to find them