I can only confidently answer the first two questions. I think Questions 3 and 4 would be too difficult to answer conclusively because the pre-Christian Anglo-Saxons didn't keep written records of North Sea correspondence or failed Irish colonisation.

It's important to accept that British culture varied considerably throughout the island by the time the Anglo-Saxons arrived. The south of England had been apart from the rest of Britain even when the Romans invaded, because firstly they were heavily influenced by the La Tene culture [1] (i.e. Continental Celtic culture) due to their strong trade links with the Celts in modern France and the Low Countries. This is evidenced by the presence of the Belgae, a tribe thought to be Belgic in origin, in South-Eastern England. Furthermore, the South was better positioned to trade with Rome, meaning that Roman commerce reached tribes like the Atrebates, Regnenses and Iceni long before the Roman Invasions of Caesar and Claudius. When the Romans arrived, they found a receptive cultural elite ready to become Client Kings, who were eventually assimilated. Indeed, Claudius' invasion was ostensibly a requested intervention by a pro-Roman king of the Regnenses, who had the Romans build him a magnificent villa at Fishbourne. It wasn't "peaceful" per se, as shown by Boudicaa's revolt, but people forget that her tribe, the Iceni, had been among the most Romanised before diplomatic ties broke down.

The North was a different story: people often use Hadrian's Wall as a neat point to mark Roman territory, but in reality the North was under tenuous Roman control. Initially the queen of the northern tribe the Brigantes submitted to Rome, but even then she was almost completely autonomous, unlike in the South where Roman Military presence was initially heavy. It is telling that during Boudicaa's revolt, Paulinus was forced to retreat North and chose to offer open, suicidal battle rather than go deep into Brigantes territory. While Tacitus, a proud Roman, would not say that Paulinus was scared of going into the wild North, instead having him shield the escape of the Roman refugees into Wales, it is clear that the Brigantes were not Romanised. And indeed, the pro-Roman queen was eventually removed and the Brigantes became openly hostile to Rome: there is no reason to doubt Tactitus' reliability when he writes how it took decades to fully pacify the Brigantes. Eventually they were conquered, but Roman grasp was always tenuous. It is perhaps telling that while the South was the economic centre of Britain, with Londinium (London) and Camulodunum (Colchester) growing up as major trade centres, Eboracum (York) remained essentially a stop off for the Legions on their way to the wall. This disparity was exasperated by the fact that Cumbria and the Pennines in the north were not suitable for profitable agriculture, whereas the South was. This not only helped foster a Romanised Southern Elite, but also allowed more Legionaries to be granted estates in the South than the North. When Legions left the North, so too did any reason to remain in towns [2] , unlike in the South where towns were the economic powerhouses.

This meant that by the point the Anglo-Saxon's arrived, in the absence of Roman rule the North quickly threw off Roman culture and returned to Tribal Structure (as shown in such poetry as Y Gododdin) while the South had become that vague term "Romano-British": they remained in Roman urban centres for a rather long time, the Roman style economic system continued (albeit minus a standard currency) and historians like Rogers [3] even argue that the South experienced prosperity in the brief period between the break with Rome and their "conquest" by Anglo-Saxons, in the absence the oppressive Roman Tax system.

But the Souths integral place in the North European trade map worked against it: it was the first port of call for the Anglo-Saxons in their settlement. While the relationship between Vortigern and Hengest+Horsa is too mythologised to be considered historical fact, it is clear that the Romano-British establishment hired large amounts of Anglo-Saxons as quasi-Foederati to defend against the Picts* and other Anglo-Saxons. This meant that Anglo-Saxons may have arrived in greater numbers in the South, the site of the infamous "Saxon Shore", because it was richer and had more Romans to defend, as shown by early settlements at Thanet. The Northern tribes operated autonomously in their defence, because they had no real urban centres to need to defend against Saxon raiders. So Anglo-Saxons arrived in the South seeking Mercenary work, so their families arrived, they settled, they intermingled, they gradually established cultural hegemony. But while the past image of a pagan horde arriving to conquer Britain might be exaggerated, there cannot have been no conflict. Indeed, the Romano-British resistance seemed to have had a decidedly Roman penchant for open warfare. The historical record is filled with real resistance figures such as Ambrosius Aurelianus (attested by Bede, Gildas and Nennius) and not so real ones (Natanleod, the British nemesis of the Saxon Cerdic in the 508AD Entry of the Anglo-Saxon Chronicle, actually is an example of folk etymology to explain the name of the village of Netley Marsh, although there was likely a pitched British vs Saxon battle at the nearby Iron-Age hill-fort of Tatchbury Mount). Wars almost certainly occurred: why else would there be such unprecedented waves of exclusively British migration to Wales, Spain and Brittany? The only other explanations could be famine or disease, and why would the Anglo-Saxons want to migrate to a land ravaged by disease and famine, unless of course they themselves created these conditions through warfare. There is increasing tendency, even on this sub, to view Anglo-Saxon settlement as one of peace, integration and assimilation. But while earlier mental images of a barbarian horde invading Britain are ridiculous, it is also ridiculous to view Britain as somehow exempt from the Europe-wide and incredibly violent Migration period, and the "cannonball" effect that Terry Jones and Alan Ereira refer to in Barbarians (i.e. one group, in this case the Anglo-Saxons, migrate due to a combination of push and pull factors into the lands of another group, in this case the Romano-British, who then migrate into the lands of another group, etc). In the North however, the early Northumbrian kingdoms of Bernicia and Deira initially hugged the East Coast, suggesting a sustained resistance by British Tribes in the interior (by the time of Alfred the Great 400 years after this period, Cumbria on the West Coast was almost completely autonomous of its nominal ruler in York), in contrast to the rapid inroads the Anglo-Saxons made in establishing kingdoms in the South while up against the small armies of the Romanised, and frankly, Soft Southerners.

This is where I can touch on your second question. Anglo-Saxon society was a slave-based society, much like the Romans. As such, historians such as Alcock [4] have identified in burials Alpha, Beta and Gamma tiers: Alpha being the elite, the Eorls/Earls and Thegns(certainly Germanic noble families), Beta being the Churls/Carls (probably Germanic warriors or high-status British "collaborators", poor landowners) and Gamma being the Gebur and ðeow (Slaves, probably mainly subjugated British people). In Southern England the ratio of Alpha to Beta to Gamma was 1:3:6, meaning there was a slim majority of British, but with unquestioned Anglo-Saxon cultural dominance. It is important to recognise that Genetics is not synonymous with Culture, and that in the South many British nobles might have adopted Anglo-Saxon culture, making archaeological assessment of numbers very hard. But there are so few Alpha burials in the North that Alcock was lead to believe that there was "a small Anglian élite over a large, indigenous population, continuing the Celtic social system of the Sub-Roman Period".

There are problems with such a conclusion being drawn from Archaeology, such as the relative poverty of the North making Alpha burials naturally less common, but this evidence combined with the fact that Northumbrian expansion inland was initially difficult leads me to this conclusion. In the south, rapid military conquest combined with large numbers of Anglo-Saxon migrants allowed the Anglo-Saxons to quickly assume a position of Cultural Hegemony. Any vestiges of Romano-British culture were either assimilated through a small British nobility and a larger British slave class, or driven out as refugees to Wales, Brittany, etc. In the North, a strong Celtic tribal opposition had re-emerged after Roman withdrawal due to a lack of any meaningful Roman civilian prescience. Therefore they offered stronger, significantly more sustained resistance to the new settlers than was offered by the Romanised south, which while perhaps not crumbling economically or socially as far as was previously assumed, struggled to fill the military void left by Legion withdrawal, and so turned to Anglo-Saxon mercenaries to defend themselves from Anglo-Saxon raiders, which went about as well as you'd expect. The relative wealth of the North and South also probably lead to a greater number of migrants to the South, making Northern Settlement harder. This is why when Anglo-Saxon culture eventually assimiliated the Romano-British and Celts in Northumbria and the Southern Kingdoms, the cultures they created were so different. The Vikings influenced the North/South divide in England substantially, but it was already there when they got to England.

Sorry for rambling, but I feel that this is a complex period, and even this is rather simplified.

*I presume, considering that your question is centred around Anglo-Saxon migration, that you don't want me to get too bogged down with the Picts