1) Why did Charles VII not come to Joan of Arc's aid when she was being tried?
This one of the great mysteries of Saint Joan's story. There may speculations but nothing conclusive. Perhaps he was glad to see her 'out of his hair' because he did not like being told what to do by a peasant. Maybe he believed that she was a witch and did deserve to die. Of course if that was true than he knew that his coronation was a lie and that he really was not King of France. So I don't think that was the answer. Maybe he believed Joan of Arc was becoming more powerful and influential than he and he was jealous of her. Maybe he believed Joan was being punished by God for her sins.
Any way, there was a Bishop at the time who wrote Charles urging him to do everything in his power to help and ransom Joan of Arc from the enemy or he would be guilty of "Monstrous Ingratitude." The King ignored the Bishop's letter.
2) Were the people who captured her afraid of her power?
Most definitely, YES. Right from the beginning the English believe she was a witch. The English captain at Orleans, Glasdale, told her that if the English ever got a hold of her they would burn her as a witch.
3) How was Joan of Arc brought up?
Joan of Arc's up bringing was no different from any of the other girls in the village. She learned her faith from her mother as well as to spin and sew, cook and clean - all the usual duties of woman's state in life. At Joan's retrial, some 25 years later the her playmates would say, "Joan of Arc was like all the other children of the village."
4)Is it true that the military commander, who gave Joan of Arc her escort to Charles was her Uncle?
No that is not true. Sir Robert de Baudricourt, the Governor of Vaucouleurs, had no family connections with Joan. Her "Uncle" Durand Laxart was one of the first people to believe in her mission. He took her to Vaucouleurs to see the Squire de Baudricourt and when Joan of Arc wanted to leave for Chinon before receiving official permission from the Governor, it was her "Uncle" and his friend, who went with her until Joan decided to turn back to Vaucouleurs and wait for the official permission that she needed.
5) How long was the battle between Joan of Arc's troops and the English?
Joan arrived in Orleans on April 29 and Orleans was delivered from English attack by the eight of May, a total of 10 days. If Joan of Arc had been given a free hand by the powers behind the French army and the army's Captains. She would have delivered Orleans in ONE DAY But the Captains and high officials did not believe in her or her mission so they would not allow her to go directly against the English as she was instructed to do by her Voices. That is why it took ten days. It took Joan only six days to drive the English forever out of the Loire River Valley.
6) How did they come to win the war so quickly?
I would not use the word "WAR" but instead use the word battles in your question. So, 'how did they come to win the battles so quickly?' Because God was helping them.
7) Why did they decide to re-trial her trial, was it just to clear her name?
They retried Joan's case because the war was now over and the people of Europe, that is the other Kings of Europe, were focusing in the fact that Charles obtained his throne and crown by the help of a convicted witch and heretic. As I said before, this idea that Charles was helped by a witch and a heretic to obtain his throne would null and void his coronation and he would not be worthy to rule France. So to put it bluntly Joan of Arc's trial was not reopened for her sake but for King Charles' sake, that he might retain his crown and power.
8) What was Joan's leadership style, communication skills, and management ability?
Joan's style of management and leadership was hands-on. She led by example. She was always in the front lines, encouraging and guiding her troops. In contrast the normal way captains led their troops was to supervise their movement from the rear.
Most of the time in battle, she carried her standard and was a non-combatant. She did not want to kill anyone. Her job was to lead, guide, and direct them. However once in the battle before Paris, she fought a Burgundian knight with her sword and captured him.
After her death, the captains with whom she fought remembered her as an able leader with great skill in military tactics of a twenty year veteran.
During her trial, she was able to hold her own against the learned, university graduate churchmen for five months.