I have Chinese Elm kept indoor apartment window in NYC with temperatures probably around high 50-60 degree. Will it go into dormancy period at these temperatures? If so less watering and lighting requirements?

I am very sorry, but it should be outdoors, even in winter.
Inside it will struggle, will be unhappy and eventually probably die.
Indoors there is no air movement, the heated air is dry, and it will not go into full dormancy; all three are big problemas for bonsai.

While I do keep my elms outdoors, where all trees belong, I'm not sure if I fully agree with Clicio.

Chinese elms are semi-deciduous. This means they'll be deciduous in a cold climate, but they won't in a warmer climate.
It depends on where the tree came from and what it's adapted to. If it comes from a cold climate, it will require a cold period and dormancy.

You have got one thing reversed: dormancy will not make a tree require less light, less light will induce dormancy.

A 10 degrees (Celsius. Most people in the world don't use these archaic units like Fahrenheit), it should be OK. If it drops its leaves, less water will be needed - but you'll see what it requires by checking the soil.

Auk is right, but my experience in Brazil is Elms kept outside in the winter are deciduous and go dormant, at least the 3 Elms I have at the moment. But yes, could be they came from a colder country. On tje other hand, they like the warm weather in spring and fall, and go dormant again in high summer.