Yes, a fixed-mag firearm with other evil features is legal, as our laws (not really a law, but an AG guideline/directive to prosecutors) define assault firearms as having a detachable magazine.
At this point, anything legal to possess in NY is legal to possess in NJ. Be aware, however, that without possession of a NJ Firearms ID card, you may only travel directly to the range, then back home. You can't go to your buddy's house to pick him up, you can't stop for coffee on the way, you shouldn't even stop for fuel. The way our law currently exists, any deviation from the most direct route between home and range is up to a judge to decide if it was "reasonably necessary under the circumstances". Don't give them the opportunity to judge...
Obviously, the firearm must be transported unloaded, and either contained in a closed and fastened case, or locked in the trunk. The case itself does not need to be locked.