Big Bang... not much to add really. We have pretty strong evidence to suggest it did happen. I'm assuming by "world", you really mean "universe", as that is what is up for debate.

The world started as a large stream of hot gasses which slowly, from the force of gravity, came together into a solid, dense sphere. As rock just barely began to form on the surface, a large, Mars-sized asteroid collided with the Earth. The resulting collision splashed a huge amount of the Earth's mass and the asteroid's mass outward and into a totally different gravitational pull. This material soon formed the moon. About a billion years later, the Earth began to form an atmosphere due to volcanoes spewing excess gasses out which was then getting caught by the gravity of the new planet.

That's how the world started.

Well the world didn't really start with the big bang, but the universe as we know it did and our world came out of the resultant events. There is really no evidence of any gods involved. Introducing a god or gods to explain things is only things more complex and inserting an answer to have an illusion of certainty.

If I wanted to be philosophical I might say "The world didn't start, the world is", however I think the most proper response is "We don't know yet". There are enough theories with the big bounce, cyclic universes, non-linear time and parallel universes that it might be practically impossible to know when or even if the world began.

Does anybody still believe it had anything to do with God?

Since no human being was around, the logical answer is, "I don't know."