I don't think this question can be adequately addressed in a single Reddit post, because New York City was a large city full of different people of different experiences, and the "late 19th Century" is a rather broad time frame. The answer is going to be pretty different if you're talking about 1851 or 1899, and if you're talking about middle class white native New Yorkers, or middle class black native New Yorkers (a small group, but they did exist by the end of the century), or immigrants from Italy, or Jewish immigrants, then the answer is going to be pretty different, too.

A good starting point to learn about this is in the Pulitzer Prize-winning history book Gotham: A History of New York City to 1898 by Edwin G. Burrows and Mike Wallace. It's over 1300 pages long, and parts 4 and 5, spanning the last 600-odd pages of the book, cover the period of 1844 to 1898 and will answer a lot of these questions. Among the chapter titles that would be of most interest include:

Part 4 Chapter 5: "The Professional-Managerial Class: The middle class expands in size..."

Part 4 Chapter 6: "Eight Hours for What We Will: The laboring classes at work, at home, at play..."

Part 4 Chapter 8: "Work or Bread!: The boom collapses in 1873, pitching the city into long-lived depression..."

Part 5 Chapter 3: "The New Immigrants: Jews, Italians, Chinese"

Part 5 Chapter 4: "That’s Entertainment!: The Broadway stage, Pulitzer, vaudeville, Tin Pan Alley, boxing, baseball, Coney Island."

Part 5 Chapter 5: "Purity Crusade: Henry George militancy and burgeoning immigrant quarters rouse middle-class reporters, writers, ministers..."

To answer some of your questions:

Did they eat in restaurants?

Yes, I wrote a post in this sub not too long ago detailing the dining options of New York and Brooklyn in 1870. Among the available cuisines at restaurants at that time were American, English, French, German, Italian, and to a lesser extent, Jewish and Chinese (but by the end of the century, those both would have been easier to find than in 1870). "American" food at the time wouldn't have included stuff like pizza, hamburgers, or hot dogs, but it would have included stuff like pot roast, sausages, roasted chicken or turkey, meat pies, and fish. Oysters were particularly popular in New York City and oyster houses were found on just about every corner.

There was also a lot of street food available, everything from peanuts and popcorn to candy, ice cream, soda, waffles, and fruit.

What were popular neighborhoods in which to live and to work?

This depends on when you're talking about, because Manhattan was aggressively expanding northward during the second half of the 19th Century. Up until the 1880s or so, most of the jobs were located downtown, while above Houston Street to Central Park was residential, and then above Central Park, it remained fairly rural. But that continuously changed as the streetcars were put in, and businesses expanded northward. The East Side was generally the more affordable side of the city to live on, while the West Side was ritzier. The Lower East Side was a hotbed of immigrants, with Chinatown being filled with Chinese immigrants, Little Italy with Italian immigrants, the Bowery with German immigrants, and later in the century, Jews and Eastern Europeans began to populate the Upper East Side and Yorkville. Harlem was rural until about the 1870s or 80s as it was built into residential housing, and then it became a destination for black Americans, many of them from New York, but also from the South who came during the Great Migration in the decades after the Civil War.

Was there a significant population of non-white middle class New Yorkers?

Yes, though it depends on your definition of "significant" and "middle class". The middle class of the late 1800s didn't always have the same opportunities as middle class people of today, and there weren't the same kinds of social programs today, like Social Security and Medicare, and medicine was rather lacking, so even among a typical "middle class" family, they'd often be housing grandparents who could no longer take care of themselves, and children suffering from diseases and health issues that made it impossible for them to go to school, and these kinds of obligations often ate family resources--and, consequently, sick children were often shuffled off to the orphanage because the family didn't have the time and resources to take care of them. This was true among native-born white Americans, as well as immigrants, as well as native-born black Americans. "Middle class" was "working class" back then, very "blue collar", and access to education beyond high school (New York did have one of the first and most comprehensive public school systems in the country, if not the world) was minimal. College was really only accessible to the upper class elite for the most part, though it wasn't impossible for an accomplished student to go into a trade school or to a two- or four-year school to get a degree to become a teacher or a civil servant or something along those lines.

What were typical jobs? Did women work?

Yes, women did work, though if the family had the means, then the women stayed home with the children while they were school-aged, but before and after that, among middle class families it wasn't uncommon for a wife to work.

To get an idea of what kinds of jobs people had at that time, you might want to flip through one of the city directories of New York City from the era. Just from page 19 (Abbott through Aberle) of the 1887 city directory, the jobs you see listed are: auctioneer, lawyer, finisher (of furniture?), clerk (probably at a store), carpenter, driver, cigar-maker, shoer (of horses?), weaver, agent, painter (probably of houses and buildings), dyer, piano-tuner, broker, rubber, dockmaster, iron-worker, importer, piano-maker, meat (probably working in some kind of slaughterhouse), furniture-maker, merchant, dressmaker, laundry, framer (probably in building construction), patterns, shoemaker, organ-builder, butcher, provisions, varnisher, "widow" (probably past working age), laborer, millwright, "rags", watch-maker, tailor, butter, coffee, platedware, easels, leather-worker, stoves, hardware, liquor, and collector (probably tax collector). Looking through more pages, you'll assuredly find many more occupations.

How frequently did they leave town?

Probably not often, but by the late 1800s, they could. This was the time when Coney Island became a weekend getaway destination for working class people. City Island in the Bronx became another destination, and boating excursions in the South Bay and in Long Island Sound were also aimed at middle class people.

Further out of town, it was possible to leave town, if you had some reason to go, though just for a weekend, you probably weren't going to get very far because railroad transportation wasn't all that fast. But you could get anywhere you wanted in New York state and New England from Grand Central Station. To go west, you still had to take a ferry to Exchange Place in Jersey City where the Pennsylvania Railroad terminated; Penn Station wouldn't be finished until 1910, with its connecting tunnel under the Hudson River.

But if you had a job out of town, or family you wanted to visit and had the means and time away from work, it was certainly possible. How probable is another story, because the job benefits in those days weren't always so great. Vacation time apart from isolated holidays wasn't really offered yet, and many jobs were still six days a week. The movement toward five day work weeks of eight hours per day were being fought for by the working class during this period.