That's a good question, now I cannot fully answer the first part of your question, but I can do my best on the linguistic questions you ask and their history. The simple question is yes, there were enough similarities for the different speakers to understand each other to a certain degree, there would be issues with understanding, but the different people would be able to communicate. A modern example of this would be Eddie Izzard using Old English to buy a cow from a modern frisian farmer The proof of this can be found in the linguistic change Old Norse (The language of the Viking raiders and settlers) brought to Old English (The language spoken by the people living there at the time) The effect were deep and one cannot today speak English without using Old Norse borrowings. Examples here would be words such as Law, egg and the Personal pronoun I. This change did not happen overnight however, it took time and shows how the relationship developed.

The first mention of Danish/Viking people in England comes from the 787 entry in Annal D of the Anglo Saxon chronicle where we learn that three ships of the"Norðmanna of Hæreðalande" arrived seemingly to do trade. So the history of contact started early and lasted beyond the Norman conquest of 1066. When it comes to the Dates it is important to note that the major movements of the Anglo Saxon migration to England happened in the rough period 400-600CE, which puts their emigration earlier than the "Viking interactions from the late 700s. It is also important to note that the British Isles were visited by Norse speakers from modern day Denmark, Norway and Iceland and elsewhere in the period most of which would have spoken a seemingly similar language to an outsider, so it would be tricky for the settlers to immediately recognize them as their past neighbors. This prolonged contact did lead to a kind of peace and cooperation between the people in question which ended in the forming of the Danelaw in the 9th century. This partial stabilization of the country also lead to increased immigration and settlement of "vikings" across England which lead to the different people intermixing further, and by the end of the Danelaw there would be little of note separating the settlers from the locals. The extent of the integration can be seen in the St. Brice's Day massacre, where the number of Norse settlers living in the land were great enough to make the idea of massacring them the only valid option in staving of a perceived coup of the English crown.

So to summarize again, linguistically they would have understood each other with some difficulty, but the differences would have quickly disappeared as the integration of the new settlers continued further eventually leading to its erasure in the 11th century.

Abels, Richard (2018) Æthelred the Unready: The Failed King. London: Allen Lane.

Jorgensen, Alice (2010) ‘Introduction: Reading the Anglo-Saxon Chronicle.’ In: Jorgensen Alice (ed.) Reading the Anglo-Saxon chronicle: language, literature, history. Turnhout: Brepols. 1-31.

Keynes, Simon. (1997) ‘The Vikings in England, c. 790-1016.’ In: Peter Sawyer (ed.) The Oxford Illustrated History of the Vikings. Oxford: Oxford University Press. 48-72.

Lavelle, Ryan (2017) Cnut: The North Sea King. London: Allen Lane.

Pons-Sanz, Sara, M. (2010) ‘Norse-Derived Vocabulary in the Anglo-Saxon Chronicle.’ In: Jorgensen Alice (ed.) Reading the Anglo-Saxon chronicle: language, literature, history. Turnhout: Brepols. 275-305.

Swanton, Michael (2000) The Anglo Saxon Chronicles. Trans. 2nd edn. London : Phoenix.

Sawyer, P, H (1981) Kings and Vikings: Scandinavia and Europe AD 700-1100 London: Methuen

Cavill, Paul (2001) Vikings: Fear and Faith. United Kingdom: Zondervan.

I wrote some short answers to the similar questions to those of OP before:

• Did the Norse have peaceful (perhaps mercantile?) contact with Britain and France before Viking raids began? If not, how did the Vikings know where to raid?
• In the TV show Vikings. It is shown that the norse and saxons in england could not understand each other. Given how close the languages are how much mutabilty was there?

I rather doubt OP's another premise that Anglo-Saxons in the 8th century had solely constituted of the descendants of the immigrant from Jutland Peninsula as well as Northern Germany, and that they inherited the shared cultural heritage from their ancestors, thus had much in common with the Norse raiders in the 9th century in the light of up-to-date academic discussions.

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I also recommend to check a few detailed answers of the flaired users in the following recent question threads:

• In the early Middle Ages English supplanted Celtic and Latin as England's main language. If you lived in Britain between 450 and 700 what would the transition look like to you?: /u/BRIStoneman
• What is the consensus on professor Susan Oosthuizen’s Contention that the Anglo Saxon invasion was probably a myth?: /u/Libertat and /u/BRIStoneman
• Why is Norse mythology so well-known compared to other contemporaneous Germanic religions?: /u/Platypuskeeper and /u/y_sengaku
• Norse Mythology survival vs Continental Germanic and Anglo-Saxon mythology: /u/Platypuskeeper

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Put it shortly:

• Recent researches regard the formative process of the Old English and the polities of the Anglo-Saxons from the 5th to the 8th century as much more complex than generally assumed (that ultimately based on the 19th century mass migration and almost total replacement (added:) hypothesis of the 'people' of the Roman-Britons with the Anglo-Saxons.
• Now the scholars admit that we know much less about Norse mythology with ease, and further, much and much less about the religious traditions among the Anglo-Saxons before their 'official-political' conversion from ca. 600 onward. Even some Christian practices, inherited from the ex-Roman local population, might have been co-existed with some non-Christian one.

Add. Reference:

• Oosthuizen, Susan. The Emergence of the English. Kalamazoo, MI: ARC Humaniries Pr., 2019.

(Edited): some typos.

We have to be very careful talking about cultural continuity across pre-Christian Germany and Scandinavia, particularly in relation to gods and mythology. Our 'knowledge' of specific pantheon and the structures of belief and worship are quite patchy, and based largely on sparse archaeological evidence and literary sources that post-date conversion by up to several centuries. /u/Platypuskeeper has written a couple of really good answers on this recently.

Often this question gets asked from the other perspective; that is whether the Anglo-Saxons were aware of the Germanic origins of the Vikings. The answer to that is that they very much were aware of their geographical origins, but took great pains to erase any ideas of cultural similarities that might have reared their head. For most of their appearances in the Anglo-Saxon Chronicle, references are simply made to 'the Pagans', 'the Enemy' or just 'the Army' from the 860s onwards. It's only later that the Danes begin to be identified as 'The Danes'. In Asser's Vita Ælfredi, the distinction is even more stark. The Danes appear ever only as 'the Heathens'. The reasons for this are readily apparent; both the Anglo-Saxon Chronicle and the Vita Ælfredi are more teleological works of family biography more than they are true histories. The battle of the brave, Christian Cerdicings against the barbarous heathen horde who seek only to rampage and despoil is a much more heroic tale than a simple Danish invasion, and one which compares well with later Chansons de Geste. Whether the invaders in these narratives were Danish, Swedish or Norwegian, or what there their actual cultural practices were are moot to the greater moral message.

The English did recognise a cultural heritage when it was convenient. When his sister Eadgyth married the ascendant Holy Roman Emperor Otto I, Æthelstan's wedding gift - a highly ornate ivory casket and ivory-covered books - makes reference to the unification of Old and New Saxons. Indeed, the link to the 'Old Saxons' is one that appears throughout the Anglo-Saxon Chronicle, which may in itself be a conscious and deliberate attempt to identify with the Christian Saxons over the at-this-point-still-Pagan Danes.

The level of integration into English society by Danes, not just in the Danelaw, suggests that the two must have been mutually intelligible. Material culture, at least into the 11th Century, remains noticeably distinct in many ways (although there was a surprising deal of disparity in 'English' material cultures until the 10th Century), but it's worth noting that this doesn't mean that there was no cross-cultural exchange. Indeed the opposite is true, and from the mid-9th century, we start seeing the appearance of West Saxon 'Trewhiddle' metalwork designs in Northumbrian Danelaw areas and indeed in Scandinavia itself, while in return Norse 'Borre' metalwork becomes part of English fashions, particularly in the Midlands.

In the 11th Century, there are apparently still clear cultural or ethnic differences between communities even after half a century of peace and integration. In a charter rebuilding an Oxford church destroyed in the St Brice's Day Massacre, Æthelred II explains the necessity of eradicating the Danes 'who spring up like weeds' across English communities.

What about the Normans? Weren't they Vikings too?