-some even say it as one syllable "ORNJ" (however, my friend from Rochester, Minnesota says "ARNJ"! Katy?? What do you hear?)
We say it with a long O. In fact, the Minnesota accent is best known for its extended, heavy handed O's, and maybe its hard R's, though they're not as hard as they are in Texas or Georgia. I don't think of myself as having an accent, but when I lived in New Orleans, some people thought I did, at least when saying O's. In fact, I myself have noticed people from the East Coast tend to say it more like "oeuw." Not in "orange," though. More like in "boat." Boeuwt.
I think I say "orange" pretty much like Eminem, although he does have a fairly pronounced Detroit accent.
And I agree with you about rhymes. They have to start with a different letter and match after that. "Door hinge" could work in a song, though, particularly a rap song (if they had some weird reason to mention a door hinge in the first place). They do that kind of thing a lot -- kind of slide through with close-enough rhymes.