The YMCA is not permanent housing. It is a hostel which is similar to a dormitory. It is an inexpensive temporary residence (inexpensive compared to Midtown hotels) for travelers.
Housing as in short term accommodation? Then yes! I'm not sure they have a limit to your stay as long as you are paying but it is a basic facility similar to dorms with shared bathrooms. Other alternatives could be long term house sitting. There are several sites for that or airbnb have long stay options depending on host.
Not possible. I recommend Airbnb on Jersey, Brooklyn or Long Island?
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