In "The Devil and Tom Walker", what details refer to the devil's special dealings in the new world?
You are right in indicating in your question that Washington Irving in this story re-writes this archetype of man-trades-soul-with-devil to give it a unique American twist, focussing on man's involvement in the New World of America. The Devil himself tells us about his involvement in the New World when Tom meets him and he asks the Devil who he is. Note how the Devil responds:
"Oh, I go by various names. I am the wild huntsman in some countries; the black miner in others. In this neighbourhood I am known by the name of the black woodsman. I am he to whom the red men consecrated this spot, and in honour of whom they now and then roasted a white man, by way of sweet-smelling sacrifice. Since the red men have been exterminated by you white savages, I amuse myself by presiding at the persecutions of Quakers and Anabaptists; I am the great patron and prompter of slave dealers, and the grand master of the Salem witches."
This of course gives Tom Walker sufficient information to be able to identify his companion as "Old Scratch", but it is also important to note how Irving is using this description of the Devil's activities to talk about the religious intolerance and slavery of his own day and present harsh criticism of these institutions.
It would have helped if you'd referred to "The Devil and Tom Walker". However, the eagle-eyed Sherlock Holmes figured it out.
Notice when this story is set: 1727, almost 50 years before the Declaration of Independence, when America was pretty much wilderness, inhabited by Indians and possibly anything else you wanted to imagine. This was the "New World", named so after Columbus and others discovered this unsuspected land across the sea.
This should give you a hint on how to find mentions of the Devil's dealings in the new world.
But the question, as you posed it, is about "special dealings" in the new world.
Irving's devil was not quite the same character as the Devil of the old world (Europe), and Irving altered some European stories to suit his purpose. Perhaps the "special dealiings" were the ones Irving made up?
See the reference link below for more information.