Well I'm not sure I would go that far when I say magic is connected to emotions.
I'm more prone to believing that Solas gathered all he did about the other companions (except for Cole) because of his long life and good observation skills, not some innate 'emotion sensing'. I mean maybe that was a thing when Arlathan was around (don't quote me on that though), but i'm not sure if it would carry into modern Thedas. Even for Solas.
Magic in Thedas = Fade. And we know that the Fade is intrinsically linked to emotions and consciousness, since if we cut a person from the Fade they lose their ability to feel things and their individuality and sense of self is diminished. I'd say that it's as strong evidence for connection between magic and emotions as you can get - and that's not even mentioning the fact that Fade's natural inhabitants reflect the minds of the living and emotions and states/feelings/aspects that come with it: Pride, Compassion, Wisdom, Faith, etc.
As for Solas and his 'long life observation skills'... but being able to observe emotions on more than just physical level would be a skill as well, just not one that many can practice today. Even the propaganda piece he's the target of mentions that he's someone who "knows much of the People and their spirits", suggesting that even with 'innate emotion sensing' it's not as simple as looking into someone's head. Cole after all makes many apt observations, but there are many things he doesn't understand, since he's "just" a spirit of compassion. But someone who is able to understand nuance can get a better picture even if elves themselves ultimately can't just peer into someone's mind as quickly and easily as a spirit does.
And as much as this might have been an ability from times long past we do know that elves retain at least some of it, or some of their connection to the Fade. That's what all the poking and prodding of Sera by Solas and Cole was supposed to show - that even a 'non-elfiest' elf you get can still 'see' or sense more than a common human. Her ears are supposedly always turned to the Fade. Her ability to place an arrow in the right spot Cole compares to his ability to know where a knife should be. She gets all kinds of bizarre feelings and deja-vus in various places, to which Solas/Cole immediately reacts, if he doesn't make an observation himself (places being filled with sadness or memory, tremors in the Veil and so on), on top of those mages themselves can make.
So yeah, I'd say that they certainly something more there.
Cole is made of simplicity,these abilities depend on that and the moment he become more complex he start to lose those abilities.
The human Cole is equal to the spirit Cole.
Solas was reached in a dream by a non mage for what i recall and i'm more inclined to believe that he didn't used any 6th sense when he was talking with others just his common sense.
Urthemiel may say that one without magic is blind because he is from a lost age and understand nothing of the current world.
Huh, pretty much everything you said pretty much supports my notion. Cole becoming human and losing a great deal of his innate abilities? Only proves that by becoming human he becomes blind to the Fade. Uthremiel not understanding people who cans't see magic? Well yeah, that's sort of the point.
Also, don't take my '6th sense' comment so literally. Plus - he can speak in a dream with a non-mage or even a dwarf, but *only* when they carry his unique magic that allow him to reach him in dreams in the first place.