I'll talk about questions 1, 2, and 3.

As for 4, well I think the answer's simply that the Anglo-Saxons no longer lived on the continent. They saw themselves as people who lived on Englalond and rightly so, because they'd been there for about 600 years at least. So even though they maintained strong ties and even sent out special missions to Christianize continental peoples out of fellow-feeling, they no longer saw the continent as their home.

Back to question 2: yes the languages were mutually intelligible. Old Saxon was a low German language, as was Old English, so the languages were very close to one another in grammar. This is because the source of the migration was the lowlands, where Holland, Denmark, and northern Germany are today. Here's what Baugh and Cable's A History of the English Language says about these language groups:

In early times we distinguish as Low German tongues Old Saxon, Old Low Franconian, Old Frisian, and Old English... Old Saxon has become the essential constituent of modern Low German or Plattdeutsch; Old Low Franconian, with some mixture of Frisian and Saxon elements, is the basis of modern Dutch in the Netherlands, and Flemish in northern Belgium; and Frisian survives in the Netherland province of Friesland...

So they are very closely related languages and at least around the early to middle 9th century, we know they were mutually intelligible, and we can guess that they were probably intelligible for a decent amount of time after that. What we know comes from an interesting Old English poem called by modern editors Genesis B, which is a translation from Old Saxon into Old English. But it's not really a translation exactly; it's really more of a transliteration, since the meter and many of the idioms and some grammatical elements are not quite Old English in character, but more Old Saxon. It is clear that an Old English poet just wrote down a Saxon poem in his own dialect. It fits decently well enough, and if you can read Old English, you can read it but if you're very familiar with OE language and verse, you know there's something strange about the poem. The standard edition of Genesis B is a study called The Saxon Genesis: An Edition of the West Saxon Genesis B and the Vatican Genesis, which tells in detail the story of the discovery of this connection. One of my professors in grad school, AN Doane, edited it. The story is very interesting: the great scholar Eduard Sievers postulated the existence of an Old Saxon Genesis (which was mentioned in the Latin preface to the Heliand, the most important Old Saxon text), and it wasn't until the very end of the 19th century that the Vatican Genesis was discovered and the relationship revealed.

So this anglicized version of a Saxon poem tells us that the languages were mutually intelligible, and, maybe more importantly, that there was cultural exchange. But where was that cultural exchange centered? What kind of "community" of Saxons was there? The answer, as always in the Middle Ages, was that the community formed a little bit around Christianity and a little bit around political leadership, both of which were typically intertwined. The poems I was talking about above were conscious efforts to retell Christian stories in Old Saxon using traditional vernacular verse. It was also a part of a long battle between Charlemagne and the Saxons, after which the Saxons were subdued, and made to convert to Christianity. This was a common tactic at the time, and was spearheaded by the Carolingian kings who were basically forcing people in the countries I discussed above to convert. The great king Charlemagne had very close ties with the English, and England's best scholar, Alcuin of York, would become the head of the Palace School at Aachen, personally invited by Charlemagne. So the intellectual relationship between the Anglo-Saxon people and the Frankish court that held power in Old Saxon speaking countries was strong, and this is reflected in the sharing of the Genesis B poem. But the relationship was complicated, since Charlemagne forcibly subdued the Saxons and converted them, which was looked upon quite favorably by the Anglo-Saxons who were happy to see them converted, even if through violence, by Charlemagne, the champion of good governance and Christianity in Germany at the time.

This leads us directly to question 3: did Paganism survive? The answer is pretty much "no," but I guess it depends on how far you're willing to stretch the idea of the term paganism. Does referring to Woden in Alfred's official genealogy amount to paganism? What if Woden's just one in a long line of people who include biblical figures as well? Christianity absorbed, transformed, and sometimes didn't much change local traditional customs, including the religious customs we think of when we say paganism. All the texts we have started in a church somewhere, because that's where texts were produced and stored. So paganism as a practice didn't survive, but elements of old customs did. This is how you get the Germanic warrior Christ, or interlace pattern on a cross, or baptismal spoons from Byzantium in the Sutton Hoo ship burial. The cultures blended together.

By 1066, the English had spent about two hundred years modeling their kingship and ecclesiastical structure on the Frankish, Carolingian model. The Saxon people never had such an immensely powerful and influential political system and ecclesiastical structure like that, but that structure was active in the lands, even if under the control of another people, the Franks.

So would the Anglo-Saxons have a kind of fellow feeling and understanding that they came from the same place as the Saxons? Yes. But that relationship is filtered through hundreds of years of history by 1066 that dramatically altered the political, intellectual, and religious cultures of both peoples.