2. Thousands of books that have been written about him.
3. Josephus does not mention him in his writings.
4. Pilate, later, in retirement, asked about Jesus of Nazareth, was wondering: "Jesus... Jesus of Nazareth... No, I can't remember him." This is from "The Oxford Illustrated History of Christianity (page 23) by the scholar John McManners.
5. That's all. I think you are right: He did not exist.