Yeah, NJ is definitely the strictest state when it comes to gun control. You are correct in assuming your air rifle or BB gun is considered a firearm. And there is nothing stopping you from carrying it around your property (Although you can expect passersby to call the police if they see you walking around your yard with a rifle, even if it is an air rifle...trust me
). Discharging it? I scoured the Criminal Law and MV Handbook for this and couldn't come up with anything. So I did a few internet searches. Seems that a lot of towns have adopted Chapter 86 and modified it to suit their needs. The gist of it seems to be that there's a determined amount of feet you are permitted to fire a weapon in proximity to any structure or dwelling, unless it's your own (I saw 300 feet and 450 feet, to name a few). Silly as it sounds, I would check to see if your town has a website that lists the local code or call someone at town hall and ask.
Time was, you'd go out in the yard and shoot and no one would bother you, but in our post 9/11/Columbine world, anyone so much as sees ANY gun they poo themselves.