The type of questions you are asking are pretty big and tend to be hard to provide a coherent answer for. I think the reason is that questions about the existence of God are really several questions all rolled up in one. How do you answer several complex questions at once? There is no one brief way that I know of that is coherent and satisfying. If you can break out the underlying questions and examine them closely, it is possible to find sensible answers. A systematic argumentative discourse of the how and why is called apologetics. There is an apologetics forum here that breaks down the individual questions – maybe you’ve already seen it. There are some teachers that you could search for that I believe explain things in a coherent way; Stephen Myer, Ravi Zacharias, William Lane Craig, …, there are literally hundreds of them. Some are purely religious teachers and others are PHD’s in some field of science and happen to be Christian as well. You could search for the "Kalam Cosmological Argument", it's a good starting point.
I think most Christians come to believe because of personal experiences more than convincing facts. My education is in engineering and I have always enjoyed the science/faith conversations, the things I can break down factually or mathematically. I enjoy pondering those things but they could never cause me to accept Christ. I saw the lives of people that I am close to be transformed in positive ways by something I did not understand and I wanted what they had, that was the first step to becoming a believer for me. Belief in God and in the bible for me is faith first and it is bolstered by my belief in creation, a personal creator, eternity, duality of man and the meaning of life. I am very comfortable with my belief about all of these things and I have no trouble explaining each of them to whomever may ask; my faith is a composite of these things. It is important to know why you believe what you believe and to be able to give a reasonable defense of your beliefs.
I hope you find the answers you are looking for and I hope you seek out thoughtful discourse about whatever individual topic is foremost in your thinking. I truly believe that the God of the bible provides intellectually sound answers as well as eternal salvation.