Norway entered into the Kalmar Union (1397), which started as a personal union with Denmark, but ultimately turned into Danish rule over Norway. When Denmark relinquished Norway to Sweden in the Treaty of Kiel (1814), Greenland, Iceland and the Faroe Islands were not included. Greenland had not been significantly populated by Scandinavians since the Middle Ages by then, and the Icelanders and Faroese had developed distinct identities.

Sweden had recognized Danish sovereignty over all of Greenland since the Treaty of Lund in 1679, and had no compelling reason to change that in 1814 or since. By the early 20th century, many other countries (the USA, Britain, France, Italy, Japan) had also formally recognized Danish sovereignty over Greenland.

The only ones who ever disputed it was the then newly-independent Norway, who first demanded Svalbard (Spitsbergen) in 1919 and agreed to put the matter to the post-WWI Paris Peace Conference (ultimately resulting in the Svalbard Treaty) in return for not making any claims to Greenland.

However, by that time a lot of Norwegian sealing and whaling was going on in east Greenland, and the Norwegian government abandoned their position in the early 1920s, under pressure from these interests. This lead to the Greenland Agreement of 1924, which gave Norway whaling rights etc but did not say anything about the matter of sovereignty.

Then what happened is that in 1931, a private five-man Norwegian expedition took the area of east Greenland under occupation, under the claim it was terra nullius and declared it to be "Erik the Red's Land". This action was not initiated nor originally supported by the Norwegian government, but under pressure from parliament, they 'approved' it and formally annexed the area a month later. The radical right-wing defense minister Vidkun Quisling (of later infamy) wanted to back it up with military force, although that did not happen.

Denmark complained to the court you mention in the Hague (the predecessor to today's International Court of Justice in the same location).

Norway didn't have much of a legal leg to stand on; there's the aforementioned fact that all of Greenland had de facto and de jure been under Danish rule for most of recent history as far as almost everyone was concerned. Denmark also backed this up with statements from the representatives of the actual people of Greenland (Inuits), supporting Danish rule (the first time they'd ever been consulted on the matter!) Third, the court decided that east Greenland was not terra nullius at all. In a 12-2 vote, Norway's occupation was declared illegal and invalid.

The reasoning behind the US proposal to buy Greenland was likely Cold War strategic interests - the shortest path from the western half of Russia to the east coast of the USA passes over Greenland. The same strategic interests lead to the US establishing Thule Air Base there.

I would guess that Denmark refused for the same reasons many other countries would have; they didn't need the money, and giving up territory is not that popular. While Denmark had sold the Virgin Islands to the US during WWI, in that case there had been a greater risk of military invasion (by either Germany or the US - which was in a period of Caribbean expansion at that time - e.g. Cuba and Puerto Rico). The risk of the US starting some kind of war of aggression over Greenland against a country they'd just fought to liberate from Germany would seem rather unlikely in 1945.

Such a sale requires parliamentary approvals and other bureaucracy; before the Virgin Islands, Denmark had reached agreements on selling certain islands, although the sales never ended up being completed, and before the actual sale a referendum had to be held. So a mere offer doesn't necessarily mean so much anyway.