One of those places is Chinatown.
Luckily, I stumbled upon 12 Corners one day. It meets all of those criteria and more.
This quest for a coffee shop started when I wanted to do some research, but wanted to be near Chinatown because I wanted to do some produce shopping without wasting a ton of time on the train getting from said coffee shop to Chinatown back to Washington Heights.
My friend was visiting and also needed to get work done, wanted to get Joe's Shanghai, and also wanted latte art since she had never had latte art before.
It is located across the street from the East Broadway (F) stop. Otherwise, you can easily access it from the Grand Street station (B,D).
The location is great. It is a pretty small café, though, but has some character. Along one wall there are about 4-5 (very) small tables (2-person seats...one bench side and one chair side). Along the windows, there are about 4 bar seats on each side and on the opposite side of the shop, there are 2 larger communal tables. They have a really book café sort of feel as there is a bookshelf filled with books on one side. It is relatively quiet in the shop since I think most people do takeout.
Their drinks are relatively inexpensive, but I think the cups are pretty small. So, I guess they end up being about the normal price for a latte.
My friend and I each had a matcha latte (mine was soy). The flavor left a lot to be desired, I mean it wasn't bad, but it also wasn't super good. Then again, I have had pretty darn good matcha while I was in Japan. They don't sweeten it at all and it kind of seemed like they didn't thoroughly mix it. This may have been a situation where I should have gone with a coffee at a coffee shop, though.
Most patrons ordered lattes or iced coffees and seemed to be really pleased with their choices, so I guess that's a try for next time.
They also sell pastries and sandwich items. The pastries come from local bakeries around the city. Once they run out, they're out for the day. Their coffee is Kobrick coffee, roasted in NYC.
They offer a student discount (with student ID), a Foursquare discount, and also use the Belly Loyalty Card system.
Check out other Chinatown posts:
Best Pork Chop House: http://eatstretchexplore.blogspot.com/2014/10/nyc-chinatown-best-pork-chop-house.html
Chinatown Ice Cream Factory: http://eatstretchexplore.blogspot.com/2014/06/nyc-chinatown-chinatown-ice-cream.html
12 Corners Coffee Shop: http://eatstretchexplore.blogspot.com/2014/10/nyc-coffeeshopcafe-chinatown-12-corners.html