As a young boy, King Harald Fairhair of Norway (c. 850 - c.930) is supposed to have sent one of his young sons to be fostered or educated by King Æthelstan in England. The son, Håkon (c. 918 – c. 961), thereafter known as Adalsteinsfostre (Aðalsteinsfóstri lit. “Æthelstan-fostered”), would later return from England and claim the throne in his own right, becoming Håkon I of Norway.
Our sources, both for Harald and Håkon are fairly poor. What little there is has largely come down to us in the form of sagas written centuries after their deaths. Yet my brief “popular-history” knowledge of Håkon’s early years piqued my interest, and I’m guessing our knowledge of the workings of the Anglo-Saxon court is probably somewhat better than of the contemporary Norwegian one.
Add to that a couple of questions more specific to Håkon’s case.
I'm not expecting an answer to all of the above questions. I'd welcome any attempt to shed some light on one or more of them, however.
I was coming back to answer this question when I had more time, but /u/Steelcan909 already covered several of the important bits! I'm actually writing a dissertation chapter (and article) right now on the raising of sons and princes in later Anglo-Saxon England, and yeah, there seems to be no mention of this episode outside later saga material; and yet, a whole range of books and articles definitely take it at face value that it happened. For those interested, there are a couple of semi-academic books that DO talk about fosterage, but the more I read of them, the more wary I am. Those are Sally Crawford, Childhood in Anglo-Saxon England, and (IIRC) Paul B. Newman, Growing Up in the Middle Ages.
Like hostage-taking, though, another related topic that is MUCH better attested than fosterage is baptismal sponsorship. We see that in Alfred sponsoring the conversion of Guthrum of East Anglia (who takes the Christian name Æthelstan, coincidentally), but a whole bunch of other royal and aristocratic connections get made that way. In fact, another figure who is often included alongside Hakon as a 'foster-son' of Æthelstan is Alan II ('Twisty-beard' a.k.a. 'Barbatorta'), Duke of Brittany, who grew up at the English court during his father's viking-caused exile. What the source (the Chronicle of Nantes) actually says is that Æthelstan 'raised him from the font,' i.e. was his baptismal sponsor (god-father). I THINK - and this is currently just a guess - that older historians might have assumed or misrepresent situations like that as fosterage considering the significance of the system to both Scandinavian (i.e., Iceland) sources AND to Irish sources (including a lot of fostering in the legendary stories). And these types of god-father/god-son relationships COULD help cement political alliances: Guthrum kept the peace until his death a decade or so after his conversion under Alfred, and Alan Twistybeard later paid homage to Louis IV of France who also happens to have been raised in the same English court (under Edward the Elder and Æthelstan) during the exile of his father, Charles the Simple.
As for your other questions!
Conflict between 'pagan' and Christian groups and individuals certainly happened, but probably no more than conflicts between Christians. Æthelstan might have been the grandson of the 'great' viking-fighter Alfred, but he had no problem marrying his sister off to Sitric Caech, a probably-pagan viking leader of Northumbria; ten years later fighting a huge battle at Brunanburh against the combined forces of Olaf Guthfrithson, viking king of Dublin, and the (Christian) kings Constantin of Alba and Owain of Strathclyde. Christian aristocrats thus were not above working with 'pagans' for their own purposes. After the viking invasion of Mercia during Alfred's reign, the Chronicle says that the vikings placed a 'foolish king's thegn' named Coenwulf on the Mercian throne; based on the West Saxon origin (and his Mercian name), it is likely he was in fact a member of a rival royal bloodline who used to vikings to help him reclaim the throne. Other evidence shows that pagan Scandinavians may have been welcomed in Anglo-Saxon (and other early medieval) courts. An Old English version of a text by Orosius, composed at Alfred's court, recounts a visit by Ohthere of Halogaland, a (presumably pagan) Norwegian who told the court of his travels north around Scandinavia to the north coast of modern-day Russia and then down to the Baltic. Royal Scandinavian exiles also made their way to the Carolingian courts, including one named Harald Klak. (For more on these, see Simon Coupland, 'From poachers to gamekeepers: Scandinavian warlords and Carolingian kings,' Early Medieval Europe 7, 1998).
And finally....
As for royal/aristocratic education in the 'long tenth century,' we lack a lot of sources to know anything TOO specific. We do know a bit, though. High-status children would have had private tutors, almost certainly clerical or monastic. Otto III, for instance, the young King of the Germans and future Emperor, employed the de-vested former archbishop Gerbert of Aurillac as a tutor before eventually installing him as Pope Sylvester II. Asser's Life of Alfred, if we believe everything it says (NB: you probably shouldn't) tells us that Alfred himself was involved in teaching his children and those of his court, and that he made sure his children knew both Latin and the vernacular. There is also a quite common genre known as 'mirrors for princes' (Latin specula principum, German Fürstenspiegeln), which are handbooks specifically meant to instruct young princes in (explicitly moral) behavior. These were generally written by monastic authors, though there is one example — Dhuoda's Liber manualis —written by a mother for her son, who was possibly a hostage at the court of Charles the Bald in the mid-ninth century. The only thing, though, is that (as far as I know) there aren't any examples written for Anglo-Saxon kings specifically; on the other hand, tenth-century Anglo-Saxon kings and princes had very close relationships with Ælfric of Eynsham (a famous homilist) and Wulfstan, Archbishop of York, both of whom (as I'm arguing) wrote an awful lot about regulating royal (and masculine) behavior, so we might think of their work as similarly instructional. We should also assume that Anglo-Saxon aristocratic boys would have been trained in military matters, and in other princely pursuits like hunting, but I don't THINK we have specific evidence about that. (The will of Æthelstan Ætheling, the son of Æthelred the Unready, does give a lot of evidence of the importance of those pursuits, but by the time of his death, he was well into young adulthood.)
King Hakon I of Norway is most likely not a real historical figure. His reign is only attested in far later sources, from Iceland, and not in any contemporary primary sources in Scandinavia or England. While this in itself is not clear evidence of his non-existence, there are other much more well attested Norse rulers coming from the same area and in the same time frame, Erik Bloodaxe being the most well known, iirc he is said to have been a relative of Hakon in the saga.
This prevents a great many of your questions from being answerable since there are no attested examples at this time frame from which to draw conclusions, especially between Heathen and Christian courts. A quick word search through the copy of the Anglo-Saxon Chronicle I use has no examples of fosterage, foster, or anything similar, but many results for hostages. This is by no means an exhaustive search through a great many primary sources, but the practice doesn't seem to have been common in the Anglo-Saxon world around Hakon's time.
Practices of fosterage are however very well attested in medieval Iceland, and much of our "information" on Hakon comes from the Icelandic sagas, it is possible that the contemporary practices of Iceland were projected back into earlier times by those who composed the sagas and not an indication of contemporary practice. I cannot speak to other specific fostering examples, but I am aware that the practice became more common later on in the Medieval period, but at this time in Anglo-Saxon England it is not a hugely attested practice. This may be in part due to the fragmentary nature of the sources available from this time period.
We also need to differentiate fostering with hostages. Hostages were a ubiquitous part of court life in the Middle Ages. Hostages did live at foreign courts regularly and would have been raised and educated in them as well, but they were not there sent to other courts for the purpose of education or connection building, instead to ensure good behavior.
If you have more questions on the interplay between Heathen and Christian members of courts though, I might be able to answer some questions on that.
Further Reading:
Anglo-Saxon Chronicle, this work details, in rough sketches, the general outline of history of the Anglo-Saxon kingdoms, it cannot be relied upon on its own, but when used in conjunction with other sources it is quite useful.
The History of William Marshal is a very useful later work that talks about fosterage and hostage taking in later Medieval England
You seem to already be somewhat familiar with the Saga of Hakon the Good.
For accounts on the transition from Heathen Anglo-Saxon kingdoms to Christianity and the events around them, we are still left only with Bede's history of the English People. It does not have heathen accounts, and it is a few generations before Hakon and Alfred, but it might be useful for learning about the interplay between the two religious systems at this time.
Secondary Sources:
Flora Spiegel's article in Anglo-Saxon England titled "The Tabernacula of Gregory the Great and the Conversion of Anglo-Saxon England" might be of interest since it also talks about the transition between heathenry and Christianity
Christopher Adam's article in the collection Conversion and Identity in the Viking Age titled "Two Modes of Religiosity in Conversion-Era Scandinavia" would also help talk about the Scandinavian side of things in regards to the interplay of these two faiths in the same court.