Unfortunately many of the questions that you have are just not able to be answered. When dealing with the Viking Age there is a gross paucity of sources that detail oriented questions, like yours, rely on. We cannot know about written treaties, the bureaucracy required for conquest, or the personal relationships that monarchs cultivated quite simply because the sources that are needed to answer these questions do not exist in many cases.

Now these kinds of sources do survive in some contexts, the treaty of Alfred and Guthrum survives to the modern day, correspondence between medieval people is often an important source, and so on. However this distribution is not even across the medieval period, both chronologically and geographically. Literacy, by which I mean Latin literacy, learning, and writing, did not come to Scandinavia until the advent of Christianity and even then it was a slow process.

Coupled with this problem, there is another, perhaps more difficult issue to contend with. In the modern day we have conceptions of warfare, diplomacy, foreign relations, and so on that have not always held through history. We tend to think of "war" as being a distinct state of a society, contrasted with peace. But this simple binary does not hold for the middle ages. There were no national presses or embassies to announce declarations of war, no centralized government to bring the resources of the nation onto a war footing.

To take your questions in specific order though, will all of the above caveats in mind.

Did he inform Sweyn that they were at war

We have no way of knowing this particularly. It is likely that there was never an official declaration of war issued in writing, but that does not mean the Danes were unaware of Norwegian ambitions or attacks.

when he eventually failed, did they conduct formal peace negotiations leading to a written treaty?

Its certainly possible that this happened. I do not know of any manuscript that specifically details such an arrangement, but it IS possible that one exists and I am not aware of it because it has not been published in a language I read. However, I would be extremely surprised if such a document ever existed.

Or was there some other arrangement?

It is perfectly possible that an unwritten understanding was reached between the two parties, but there is obviously no way to record that in literary or material format.

Similarly, if he had succeeded in killing Sweyn, how would he have imposed his rule, presumably against the opposition of many important Danish landowners?

Askhistorians is not in the business of counter factuals, as they are by nature not history but speculation. However I will say, that the only way Harald would be acclaimed king of Denmark would be by those same leading figures.

How did these early kings conceptualise what we would call foreign relations, and how did they conduct them?

This is the most interesting part of your question to me personally. In one sense the answer is easy, we don't really know because there is not a great deal of surviving manuscript material to base arguments on about foreign relations. However, the bigger issue I think is to try and understand what foreign relations could mean to these people. After all we are a ways away from nation state governments with state departments, diplomatic corps, and the like. "Foreign relations" would not exist as we envision, but would instead be a much more individually driven force. We also need to keep in mind that the internal divisions that monarchs inexorably had to deal with, troublesome vassals who could appeal to outside forces, the Church, their own households, and so on.