One key element is the ethnic and cultural clashes. Different immigrant groups were vying for power and territory in the rough streets of New York. Another is the role of the corrupt political machines that often had ties to these gangs. They would use the gangs for their own political ends, like getting votes through intimidation or other underhanded means.
The Gangs of New York is mainly a work of fiction. Although it's inspired by some elements of the historical period it portrays, the specific events and characters are largely created for the movie.
Definitely not. 'Gangs of New York' is a work of fiction that incorporates some aspects of the historical context but is not a true account of specific real-life events or people.
It reflects the extreme division in society. There were the rich and powerful who lived in luxury, while the gang members and poor immigrants struggled in the slums. The back story shows how the city was a battleground for different classes and ethnic groups.