You cannot drive your Class B - mostly because of its height on some parkways in New York and Connecticut. I have not come across any in New Jersey that are a problem. In New York there are general height restrictions on anything named "Parkway" - as opposed to expressway, turnpike, highway, thruway, etc. Parkways on Long Island are posted at entrances - "No vehicles over 7'10"" The overpasses can be lower than this and without an exit ramp before you get to them. We have driven on parkways when we know for certain (having driven the road first in car) that the overpasses are high enough. This doesn't work when you are driving your Class B somewhere that you have not been. This is what an RV specific GPS is good for. You may get pulled over for being on a "Parkway" in NY but it is important to know for yourself what the height restriction is on that road. You can't drive a Class C or a Class A or any type of trailer on the parkways but if it is a conversion van and you are under the height restriction you have at least some reasoning to present if you get stopped. In NY the registration is "Passenger" and there only is Passenger or Commercial and in NY you are paying in that registration to be able to use the parkways.
The Garden State Parkway allows trucks so you are good to go on that. It is pretty safe to say that if a truck can drive on it, so can your Class B. When in doubt follow the trucks.
As to paying different tolls - we have not been charged a higher toll on the NJ Turnpike. We are almost always charged a higher toll on the PA Turnpike. The bridges may or may not charge a higher toll and we have been charged higher on the same bridge on some trips and not on others and there is no way to dispute this - they always say you do not qualify for the lower car toll. There is a bridge from NJ to PA that will charge a higher toll if you use an Easy Pass and not if you pay cash. Other bridges and roads with toll takers will charge you more when the toll taker sees that it is a Class B RV and not just a van. It really is hit and miss what you may pay - at least in the Northeast. Many of the tolls weigh the vehicle as it goes through and that determines the toll -and the Class B is in the higher weight class.
Also be aware that some tunnels and bridges do not allow propane so you must route around those.
The East Coast is where you will encounter both the restricted roads issues and the toll issues the most. Get south of D.C. and the tolls stop as do most of the restricted roads.