A Chinese 2 cycle only adds about 20 pounds to the bike. Getting bright lights will depend on the size battery you have as these engines won't supply power for a higher 3W or so light. Not in NJ..can't answer what will happen, and you probably don't want to know, I'm sure it's not good. Maybe you need to stick with a 49cc.
$160.00 limit...you want might buy off of Fleabay. They are all a crap shot. ignition key for the electrical shouldn't be any problem, that is if you can handle the wiring of it.
Good start is to read this, along with the many stickies.