Sammy - I live in Manhattan and my office is down near where you live in Middlesex County. If you're in the proper residential neighborhood, parking on the street is no big deal, especially during the holiday season when alternate side of the street parking is suspended for many days (this year from the 25 thru the 28th and again on Jan 1). In those neighborhoods it is not very difficult to find a spot in the late afternoon, about 4PM when the contractors and doormen are knocking off, in the early evening about 7PM when some of the daytime restricted spots come off prohibition and relieve the pressure, and about 1130PM when the bridge and tunnel crowd which has come into town for the evening starts heading for home. Obviously, it is no problem at all to find a spot just as the prohibitions are coming off, at either 1045AM where 8AM-11AM is street cleaning time or at 145PM where 11AM-2PM are the times.
Don't worry about your car even it is one of those popular with the car thieves. There is very little chance of anything happening unless you do something dumb like leaving something of possible value exposed in the interior of the car. I have at times left newish fancy cars with Jersey tags out there for days and nothing happened. I would certainly not expect the same thing to happen in or around Newark or New Brunswick, which are probably your frames of reference for that sort of thing.
Remember, driving in town is OK even during the holidays as long as you avoid the center core. Stay near the rivers, use the Drives, 9th and 10th and 1st and 2nd Avenues, and you'll generally have little or no problems, except in rush hours, when all bets are off.
The info above about those less expensive lots is right, but a specific answer to one of your questions is that most parking lots and garages generally charge $20-27 a day with no discounts for multiple day stays and no in-and-out privileges anyhwere unless they are part of your hotel stay and the hotel has an arrangement with the garage.