A) Did these natural sciences appear without being caused? And appear for no purpose?
Natural science appeared first with the Greeks, then the Muslims and finally the Europeans. They appeared because humans became intelligent and began to think own their own rather than believing in dogma. So there is both a cause and a purpose to why natural science was developed. But it was in spite of god, not because of god. Or did you ask the question incorrectly. That is the answer to the question you asked.
B) Did these natural sciences create or make itself?
Obviously not. Humans made natural science. Maybe we are lucky that nature works logically, except when you get too deep into it, like with GR and QM. How could a natural science make itself? That is a stupid question. It makes me think that you again asked the question incorrectly. God did not make natural science. Were you trying to ask if god created nature or mathematics? Is English your first language?
C) Did this natural science have a cause or maker outside of itself?
Obviously yes. Humans made natural science. Humans did not make nature. Humans are a product of nature. But natural science is the human understanding of how nature works. It's more accurate to say that humans discovered how nature works, at least at a superficial level, and that science is how humans describe how nature works.
With some early time exceptions regarding conditions around the big bang, science can observe via astronomy and derive via physics how the universe came to be. Astronomers also observe how stars are formed and can simulate how they cluster. They can observe that the universe expansion is accelerating.
Regarding the advent of life, and again with perhaps some exceptions around the first budding of life, earth science can observe via fossils, etc., the HISTORY of life-forms on the earth. No theory, just observations and the ability to date fossils and earth strata via physics. Evolution is a theory. But fossils are data, not theory. The causes are theory. But god is also a theory. Scripture, the data of god, is inconsistent with the history of life-forms and astronomical observation as well.
Scripture is made by humans. Earth science data and light from distant stars, and the first light of the universe, were not made by humans. Physics can derive the first light and the star light from principles that are verified by experiment. The scripture made by humans is not only inconsistent with science, it is just plain incorrect. The cause of that is that the science available to the writers of the scripture was incorrect. They just did the best they could and lied that it came from god in order for the people to believe in it.
But we do not need to lie anymore. We can prove by experiment and observation how nature works, as long as we do not go too deeply.
That does not prove that god does not exist. It does not prove that god does not meddle in the universe within the limits of natural principles.
Using these natural princples, humans can postulate (but not prove) how the universe came into existense, for example from other universes. So science has theories of creation that are just as likely as the god creation stories (not even theory). In these theories, the physics ones, there is no need for a first cause. So on that basis there is no need for a god. But there still may be a god. It's all theory.
I hope that answers the questions I think you really wanted to ask.