Speaking of "Battle Row" - I first came across the term in The Gangs of New York by Herbert Asbury, but I just found another reference to it in The Thirty-Sixth Annual Report of the New York Association for Improving the Condition of the Poor for the Year 1879 (published about the same time my people moved to the block):
On West 39th Street, between Tenth and Eleventh Avenues, stand a block of tenements known as "Battle Row," and almost equally well as "Murderer's Row." For years this place has borne an evil reputation, having always been a source of great trouble to the police. Its inmates are the terror of the neighborhood. It is the cradle of some of the worst Tenth Avenue gangs, and the scene of constant broils, both domestic and with whoever the roughs may chance to pick a quarrel. Arrests are of such frequent occurrence as to excite but little remark. The police themselves are frequently attacked, one, nicknamed " The Brute," having been knocked senseless with a brick only three or four weeks ago; another, the officer with whom I conversed, was himself struck with a similar missile....More can be found here. It's no wonder the Grays escaped!