Nonsense. People understood the world before long science was invented, and very little of our useful understanding of the world comes from science. Most of it comes from direct experience or the experience passed on to us by others.
Nonsense. People may not always use science deliberately, but using direct experience along with inductive and deductive reasoning is the bedrock of scientific discovery. The scientific method is not the only way to gain knowledge, but it is also not the only method in which people perform scientific studies.
Science exists because people invented it.
Correction: people discovered science. Inventing things is engineering; learning the concepts and natural laws that allow us to invent things is science.
[Science] does not, by any stretch of the imagination, represent the sole mechanism for understanding the world.
Yes it does. Without logical arguments there is no verifiable way to ensure you know anything. Under circumstances where too much is unknown you can use very weak arguments, but they must still be backed up by some form of logical reasoning.
Even belief in God uses deductive and inductive reasoning. It is just very flaky reasoning that allows the believer to make no meaningful predictions.