Pigen var rørt af den specielle fugle. : She was touched by the bird, in the sense that she felt it spiritually. But you need the passive. Touch (and røre) involve movement of hand toward object. Feel, føle the sensation.
That is a hard question. Sometimes questions here really make me think. Someone else may have a better explanation.
In English, touch and feel, can mean the same thing, depending on context. maybe they don't in Danish.
I'm American but I've spoken Danish for more than 30 years, have a Danish MA and taught English (and German) in gymnasier. I don't recall this being an issue. As you said, depending on circumstances, which is what I was trying to figure out. At any rate felt doesn't seem right here to translate this Danish sentence, so there is probably one of those frustrating slight differences. Sometimes you can, sometimes you can't.