England was never fully conquered by the either the danish or the vikings, most of the south east was never under the rule of Saxon kings and the vikings never gained a complete control of England, just parts of it, furthermore the idea that all of England was under a single throne at any point during the Saxon period of the Viking migratory period is incorrect as well. It wasn't until the Norman invasion that that occurred. We know for a fact that Mercia never unified and the entire area we now know as England was broken up into multiple small kingdoms.
By the danish OR the vikings? The danes were vikings too.
Danish Knut den Store(Canute), with help from the Norwegian Eirik Jarl, at that time the most skillfull strategist in the northern lands, did conquer England yes, all of what they at that time considered to be England in our tongue.
Furthermore while much of Scandinavian architectural forms have much in common with the Anglo Saxon and Germanic styles this is not uprising, as you said there was a migration north of Germain people into Scandinavia. Yet again i want to stress the point that that does NOT make Scandinavians Germans that moved north. There was NO unified Germanic culture, the term Germanic is used now to describe a genotype not a culture.
It is clearly visible in the north germanic languages that we are indeed from Germany, even i can understand parts of Frisian TODAY, just by the knowledge of my own tongue.
Same goes for the culture, there are plenty of things that bind the Germanic cultures together. That each germanian culture has diffrence, that is quite obvious, and we in Scandinavia managed to hold onto our culture longer then the others as we were further away from foreign influence. This is especially true for Iceland, mid,west and north-Norway and inland west-Sweden.
The point I'm trying to make is that the idea that barbarian style demonstrated in fantasy worlds is a viking thing is incorrect. Yes, they have much in common but they equally have just as much in common with other cultural types from Europe. Specificly art styles and architectural.
That they have been overused and sometimes adding things that have nothing to do with them(mostly the horns) i agree with. But the architecture and other things are mostly correct. As you would know if you had actually studied this. We use the same building style in Norway still today, both stavlaft and laft. Only out of the 3 building styles the viking use its grindbygg that is not used for houses anymore in Norway. But it is still used for eating-halls and various other things, aswell as recreation, a few recently put up in Romsdal by the viking organization there.
Woodcarving dragonstyle(called drageskurd) is also a old tradition in Norway, more common here then in Sweden and Denmark or Iceland, and still very much in use today, which i would know since im taking a course in it.
The biggest error hollywood and others do, when picturing vikings, is, apart from the horns, that they show us ONLY as warriors. And often too brutal or simply evil.
The movie Pathfinder is probably one of the best examples for that, piece of crap.
We were hardly any more barbaric or brutal then other people in Europe at the time. The diffrence is that the other Europeans, Charlemagne for instance, was christian, and the people who wrote the books were christians, we were not.
The barbaric view is mostly spewed out by christian monks in Britain. But the norse litterature dont do this.