How independent is an assistant professor in Denmark? In Denmark, an assistant professor is at the same level as a postdoc in terms of their salary scale. I believe assistant professors in Denmark can do independent research and can apply for their own fundings, but do they also have an obligation to work on what their managers (more senior professors) tell them to do? If so, how should the priority be?
Most Professors in Denmark are Assistant or Associate Professors (AP) not full Professors (FP). Even if you are deemed competent to become an FP, you can only get that promotion if there is such a position available. Otherwise, you remain an AP. FPs have higher salaries than APs, and they often have an extra budget for their research. They have become FPs because their topic is of strategic importance to the Dept. From that also often follow extra funds. 

Both as an AP and FP you can graduate PhDs. You can carry out independent research under the given economic limits. Very often, maybe always nowadays, you (FP or AP) are expected to seek your own funding. That's a kind of independence too; you decide what to apply for. 

Hierarchies are flat in Denmark. We don't think about who are APs or FPs (or Technicians for that matter) in professional work. Well, that's the case where I go (Natural Science). I have heard stories from Health Science where this is not so, and the Professors behave like kings (or Gods). The general Danish population scoffs at such behaviour; I believe they are a dying race.