Your take-home pay should be somewhere in the $2500/mo range. That leaves you with about $600,after paying your rent and parents mortgage, for everything else in life (utilities, cellphone, ect). That's not likely enough to save up for the things you mention.

Food: As cheap or expensive as you want. Typical lunch in the city will run $10-$15. Buy cheap groceries and bring a lunch every day.

Commute: Depends on where in NJ you are. Hopefully you are close to PATH, if not you will likely take a NJ Transit train/bus.

Reference: I am an iOS Engineer, working in Manhattan, living in Brooklyn. Have been here 5.5 yrs.