Answer to your questions:
1. What role does balance play in Daoist teaching?
Daoism looks for the basics or true nature of the world such as simplicity and disinterest in conformity with the Tao (Great Spirit) and leading a life expressing the essence of spontaneity. So in Daoist teaching balance looks for a condition in which different elements are in the correct proportions.
2. What did Confucius think was the key to a happy society?
For Confucius the key for society happiness was: Love for each other, keeping of traditions, education and be human and compassionate.
3. Who was Shi Huangdi, and why was he significant?
Shi Huangdi was the first emperor of China and founder of the Qin Dynasty
He was significant because he unified China with the first dynasty, building the Great Wall and the Terracotta Army found in his grave to protect him in his after life.
4. Why did Shi Huangdi move noble families to the capital and forbid the private ownership of arms?
Shi Huandi decided to move noble families to the capital to weaken rival nobles since nobles were somehow discontent because they were replaced with officials and were reduced in power.
He forbids the private ownership of arms to prevent any rebellions.
5. What were the advantages and disadvantages of the harsh Legalist policies that Shi Huangdi used to govern the large Qin Empire?
Advantages: Unification of China under Qin Policies; Standardization of writing, coins, weights and measures used across china and even laws were standardize.
Disadvantages: The emperor Shi Huandi ordered to burn most Confucian books so he was not well liked by Confucians besides he took all the arms to prevent any rebellions.
6. Do you think that the Han civil service examination system was a good method for choosing government officials? Why or why not?
Only wealthy people and recommendations from local officials could become government officials but they were well educated so looking it from this perspective the han civil method was a good one sine created stability for more than 2000 years.