Q. Why did God allow King Saul’s rule to continue so long after He withdrew from Saul? Also, why would King Saul’s army willingly hunt to kill an innocent man?
We don’t know precisely how long Saul ruled after God’s Spirit withdrew from him, but it does seem to have been a period of some years. Perhaps the simple answer to the question of why God allowed Saul to continue to reign is that time was required to prepare David to be a better kind of king than Saul had been. During his years serving Saul, first as a court musician and then as an army commander, and during the years when he was fleeing from Saul, David had the opportunity to gain much experience and learn many lessons that enabled him to be a better king. Unfortunately there seem to have been some lessons that David failed to learn or forgot, but overall he made Israel much more the kind of place God wanted it to be than Saul did.
This may be best illustrated by the answer to your second question. When Saul, out of jealousy, first told his son Jonathan (the crown prince) and his commanders to kill David, Jonathan defended David and so Saul agreed not to kill him. But Saul soon became jealous and murderous again. It seems that, without telling Jonathan, he wanted his commanders to kill David, but they did not cooperate. However, a foreigner named Doeg the Edomite told Saul that the priests at the city of Nob had helped David, and so Saul went there and had Doeg kill all the priests and their entire families.
This seems to have been the beginning of a reign of terror. The implication was that Saul would also kill the entire family of anyone else who helped David. (This might explain why Nabal, for example, would do nothing for David, although his bad character alone may be sufficient to explain that.) We learn later in the Bible that Saul had also killed many people from a tribe that the Israelites had sworn to leave peacefully alone. Saul did that in order to take their land.
So we can imagine that Saul’s soldiers and commanders feared for their own lives and for the safety of their families and that is why they pursued David, even though they knew that he was innocent. When a person in power is bent on doing wrong, unfortunately that leads many people who are under that person’s power to do wrong as well.
While David was guilty of his own sins against Uriah and Bathsheba through the abuse of his kingly power, he certainly did not have a reign of terror as Saul did. For the most part the Israelites under his reign were free from oppression and enjoyed a time of peace and prosperity during which they worshiped the true God. That is why the Bible uses David as the standard by which it measures all subsequent kings. The book of Kings, recorded by the prophets in Israel, puts it this way: “David had done what was right in the eyes of the Lord and had not failed to keep any of the Lord’s commands all the days of his life—except in the case of Uriah the Hittite.” And as I said before, perhaps it was to give David time to develop into this kind of king that God allowed Saul to stay on the throne for several more years.
Good Question 
Is our Christ’s real name, which he received at circumcision, Yehoshua
Yes, in Hebrew or Aramaic his name would be Yehoshua. That is what his parents and family likely called him. Jesus is the Greek version of that name.