While traveling through a state, one is supposed to be protected by FOPA [0]. An uncharitable (technically correct but against the spirit) reading of the law is used by New Jersey and New York to harass gun owners, as in this case and in other more egregious cases.
[0] https://en.wikipedia.org/wiki/Firearm_Owners_Protection_Act
To your first question, I think it's fair to say that both NJ and NYC (and because of NYC, NY to a lesser extent) are generally perceived as being hostile towards firearms and general weapons ownership. If I were to predict states that restricted possession of an inoperable or antique firearm, NJ and NY would be at the top of my list, in that order.
To your second question, the appearance of the object has no bearing on it's legal status as a firearm (laws prohibiting look-alike objects notwithstanding). A pistol frame or lower receiver, incapable of being anything but a paperweight without all the other parts, is a firearm for legal purposes and would get one busted in the same manner as if it were a completely operational firearm.
Weapons laws are nuanced, pedantic, and frustrating. Frequently, they make no rational sense.
A few years ago, Greg Revell had a layover in New Jersey and bad weather caused his flight to be delayed. He stayed in a hotel, and New Jersey wound up arresting him and confiscating his firearm which was illegal in the state.
He argued that he was still traveling; New Jersey said he wasn't. He took it to court and won.
tl;dr Traveling through a state with a firearm is legal, but law enforcement can (apparently) still harress you even if you are obeying the law.
I was watching a podcast a while back on some guys about securing baggage, it also may have been a gunnit podcast where one of the guys did this, and he registered his credit card as a firearm with the trust. All of the requirements for a firearm can also apply to a credit card (basically it has a serial), so he was able to register his "firearm" with his trust and take it through using the process for checking an actual firearm.
This is the Gunnit Live podcast from some of the mods and others that are regulars on /r/guns