More specifically:
Chinese immigrants were shown living there and owning their own businesses, had this happened already?
Was "The Bowry" really such a large structure? Did it really have such a large population living inside of it?
Had any Irish immigrants actually been elected into public office, as depicted in the film?
Did the U.S. Navy really fire directly into the city during the riots?
Were the "Dead Rabbits" a real Irish "gang"?
I know "Bill the Butcher" was based on several different real people, but were the "Nativists" a real gang, and did they really hold sway over the Five Points?
This is far from the fields I tend to be interested in, but the historical accuracy of the movie is...well...crap.
Although not an expert I can answer a few of your points directly:
The depiction of the Chinese immigrants was laughable. Although certainly some Chinese would have lived in NY by then, they didn't start to seriously move into the area until some years after the movie timeline ended. The Chinese Theater depicted in the film wouldn't be built until some 30 years after the movie timeline ends.
The Bowery is a street, not a building. The building shown in the movie is The Old Brewery and was used as a low-rent tenement, probably exagerated for the movie. Although it was around when the movie timeline starts, it would have been destroyed by the time the movie timeline ended.
The U.S. Navy never fired at crowds during the Draft Riots.
The Dead Rabbits were a real gang, mainly composed of Irish immigrants, who were known for robbery and pick-pocketing. They also frequently brawled with their sworn enemies the Bowery Boys. There is some disagreement as to whether the Dead Rabbits were an offshoot of the Roach Guard or if "dead rabbit" was merely a pejorative nickname made up by other gangs to refer to the Roach Guard.
I'm fuzzy here but I don't believe there was a gang called "The Natives" although the ideology was real. Gangs like the Bowery Boys only accepted "natives" and despised immigrants.
Hopefully someone who has put some serious study into this era will come along and expand (or correct) my little blurb here, as the topic deserves better.