Is there the same nuance in Danish as there is in English with <<Jeg læser dig avisen>> and <<Jeg læser avisen for dig>>? In English the second sentence (I read the newspaper for you) feels like the speaker is reading the newspaper as a favor or to help. Is it the same in Danish?
It's easier to look at "jeg" and "mig" first, as these are different also in English: "jeg" = English "I", "mig" = English "me".
With "du" and "dig" it works in the same way, except that English uses the same word "you" for both.
Example: "You talk with me and I talk with you" = "Du taler med mig og jeg taler med dig".