Золото́е кольцо́ is a ring of historical cities. I wondered if iron ring had a Soviet meaning
It doesn't. Except for the term "iron curtain", the adjective "железный" doesn't contain any particularly Soviet meaning.
You are right that "желез" is a stem for the words related to iron. However the actual noun that means "iron" is "железо".
I don't think so. An iron road, though, will mean railroad, when you encounter that.
Well, in Canada, engineers often wear an iron ring, though that doesn't have much to do with Russia or Russian.
It can be used also like "Он окружён тесным кольцом соратников", "журналисты взяли политика в кольцо" etc. But not iron, though
Кольцо is a flat geometric figure formed by two circles with a common center and different radii. But there is another sense of the phrase. For example,
The hauberk is a long shirt of iron rings
She barely escaped from the iron rings of his embrace
I'm assuming a Russian-Canadian engineer had something to do with this :)