Yes (mostly). The Anglo part of Anglo-Saxon (and by extension, the root of the word Anglian, and eventually England) refers to the Angles, who came from northern Germany/southern Denmark. They were straight up migrants, ending up in (surprise!) Anglia and also northern England. This is all in the aftermath of the Romans leaving Britain - about 4-500AD. After a few hundred years everyone is pretty much speaking the same language and married to each other's cousins, so we become Anglo-Saxons.

I think that you really mean the vikings, as in Viking with a capital V, as in Lindisfarne and looting and pillaging. In which case, the answer is also yes. After a short period of looting and pillaging, vikings did come over and settle, particularly in the north - about 850AD or so. This bit of England was the Danelaw. So called because this area, the northwest or so of England (and bits of Scotland too) was under the rule of the Danes, and populated by migrants from Scandinavia - they would have brought their families, goods, animals, language, customs, and religion with them. At this time, England wasn't England; it was a collection of small kingdoms. These kingdoms were frequently in conflict with the Danelaw (and each other) over boundaries and resources. The story of Alfred the Great is centred around his kingdom defeating the encroaching Danes looking to expand their new little kingdom, defining the limits of their respective territories, as well as establishing a diplomatic relationship. This was not the end, though, as they continued to take bits of each other's territory and fight battles. Religion was also a hot topic, as the Danelaw still had significant pagan elements.

Just after 1000AD, there was a Danish king of unified England, followed by several descendants (you probably heard of one, Cnut, or Canute, who tried to turn back the tide by commanding it. Supposably). This mini-dynasty didn't last, as it was Edward the Confessor who re-established Anglo-Saxon rule. Unfortunately, he left no clear heir, and this ended with the Norman Conquest in 1066. By this point there appears to be a degree of integration between the Danelaw and not-Danelaw, at least enough that they aren't fighting each other. After 1066 they have bigger things to worry about that fighting each other.

the Norse legacy is seen in language, particularly place names; the further north you go, the higher the concentration of Norse place names. Yorkshire is one of these, as York was the centre of control and settlement. They called it Jorvík.