Modern dance-is a broad genre of western concert or theatrical dance which included dance styles such as ballet, folk, ethnic, religious, and social dancing; and primarily arose out of Europe and the United States in the late 19th and early 20th centuries.
Contemporary dance- is a style of expressive dance that combines elements of several dance genres including modern, jazz, lyrical and classical ballet. Contemporary dancers strive to connect the mind and the body through fluid dance movements (hope that helps!!)
Source 1 would be the most reliable. This being due to the fact that it contains information about the most recent advances, which you need for your paper.
Answer:
It's not necessary but is appreciated because it means you care if your successful. It means you are driven to got that extra mile, it lets you go that extra step from being successful and a successful person. Competition drives people to doing stuff they never dreamed they could have so in a way it can be important.