To me, if I look at it, vikings were in a disadvantage: they were outnumbered, their supply lines were long (and could not get new men during winter), had no true allies to speak of, land was unfamiliar and local populace probably hostile. Yet they conquered much of current England and held it for a long time. How?
England at the time of the Viking invasions in the mid to late 800's was divided between several kingdoms which were unable to stop the depredations of the Norsemen until Alfred the Great defeated Guthrum some years after the original army landed. Each of the Anglo-Saxon kingdoms had their own dealings with the invaders and there was no one unified front presented to the Norsemen.
Northumbria, which is today southern Scotland and Northern England, at the time of the viking incursions was divided between two feuding rulers who attempted to unite against the vikings but failed. Mercia, occupying central England, was overrun soon afterwards. East Anglia which hugged the eastern coast of south-eastern England had early relations with the Norsemen but was also overrun. Wessex was the home of Alfred the Great and the only kingdom to survive the viking attacks. It went on to unify the various kingdoms into one kingdom of England.
So the disunity of the various Anglo-Saxon kingdoms goes some way towards describing why the situation was not as dire for the Norsemen as it might seem. Of course there are no specifics about the logistics of supplies and reinforcements, but the army seems to have been quite mobile across England, never getting bogged down in one place for too long. Anglo-Saxon efforts at resisting them were largely unsuccessful until Guthrum was defeated by Alfred the Great.
Other factors that you mention cannot be verified with any certainty. While estimates for the strength of the heathen army and the English kingdoms exist, they are not laid out for us by any reliable source, and any numbers are going to involve a certain amount of guesswork and speculation.
This also applies to factors like supply lines and reinforcements, though this would become less of an issue later on as the Danelaw was firmly established and the Norsemen came to ruler over much of England. As for the terrain being unfamiliar, the Norse had been launching raids on England for almost a century by this point, it is not unreasonable to think they had a good lay of the land. In addition to this, as I mentioned above they were not wholly arrayed against all the native Anglo-Saxons. The Anglo-Saxon chronicle mentions that the invaders struck a deal with the king of East Anglia and installed a puppet ruler in Northumbria after they conquered it.
The other part of your question, how did the Norse manage to hold onto this territory, is also a little murky in the sources. The Anglo-Saxon chronicle and other sources often just take it for granted that the Norse were in control after conquering the land from the Anglo-Saxon kingdoms.
Part of the reason they were able to stay in power for as long as they did, is that the Norse reached agreements with native rulers and maintained control over still nominally Anglo-Saxon areas, in addition to recognition of their own holdings. In the peace treaty between Guthrum and Alfred the borders of each realm are recognized and drawn out. I've already mentioned how the Norse were able to rule in Northumbria in the short term, through a puppet ruler.
Successive waves of Scandinavians would also come to England as conquerors after this time period. Most famously Sweyn Forkbeard and Cnut the Great established Danish control over all of England and were viewed as legitimate monarchs. This continued after the new milennium arrived. Following the death of Edward the Confessor, who himself only came to the throne after the Danish dynasty in charge died out, two Scandinavians still had claims on the throne, Harald Hardraade of Norway and Sweyn of Denmark.
Now if you are more interested in a purely military run down of the situation, I'm not sure exactly where to go, perhaps one of the resident medieval warfare flairs will chime in.
Sources and further reading:
For primary sources there are really two of note, the biography of Alfred the Great by Asser and the Anglo-Saxon Chronicle. Both sources are obviously tilted against the invaders however. A later Danish source, Gesta Danorum, by Saxo "Grammaticus" is of interest ebcause it details much from the Scandinavia side of things, though it is quite caught up in mythology and epic stories over real history.
Some secondary sources that are of note and would be a good place for further reading would be Gwyn Jones, A History of the Vikings as well as Peter Sawyer's Kings and Vikings
The Vikings began to raid England in the late 8th century for the simple reason that it was where the money was. England was increasingly prosperous, and had a lot of wealth concentrated in seaside towns and especially monasteries. The Vikings had the advantage of high mobility with their ships. They could land, raid and plunder, and escape before any English armed response could arrive. The English were also politically divided at this point, with the four kingdoms of Wessex, East Anglia, Northumbria, and Mercia sometimes simply paying off the Vikings to go attack someone else. The Vikings were content to continue the raids, until the Viking leader Ragnar Lothbrok was captured by the Northumbrian king Ælla, who had had quite enough of his kingdom being plundered. The commonly told account is that Ragnar was executed by being thrown in a pit of poisonous snakes.
The sons of Ragnar swore revenge on Ælla, and prepared a large invasion force. The invaders initially landed in East Anglia, whose king gave them horses to get them to go away. Here are the entries from the Anglo-Saxon Chronicle for the next two years:
A.D. 865. This year sat the heathen army in the isle of Thanet, and made peace with the men of Kent, who promised money therewith; but under the security of peace, and the promise of money, the army in the night stole up the country, and overran all Kent eastward.
A.D. 866. This year Ethered, brother of Ethelbert, took to the West-Saxon government; and the same year came a large heathen army into England, and fixed their winter-quarters in East- Anglia, where they were soon horsed; and the inhabitants made peace with them.
The Vikings went north and took the city of York. Ælla and his predecessor king Osberht led an army against the Vikings, but were defeated. Anglo-Saxon accounts say that Ælla fell in battle, while the Norse accounts say he ws captured and executed by the blood eagle. Here's the Chronicle:
A.D. 867. This year the army went from the East-Angles over the mouth of the Humber to the Northumbrians, as far as York. And there was much dissension in that nation among themselves; they had deposed their king Osbert, and had admitted Aella, who had no natural claim. Late in the year, however, they returned to their allegiance, and they were now fighting against the common enemy; having collected a vast force, with which they fought the army at York; and breaking open the town, some of them entered in. Then was there an immense slaughter of the Northumbrians, some within and some without; and both the kings were slain on the spot. The survivors made peace with the army.
Over the next several years, the Viking army conquered East Anglia and Mercia, leaving only Wessex in Anglo-Saxon hands. But Wessex was now led by Alfred the Great. After a series of close escapes battles, Alfred was able to drive the Vikings back and defeated them in the Battle of Edington in 878. An agreement was forged with the Viking leader Guthrum in which the Vikings were acknowledged to rule over East Anglia, eastern Mercia, and most of Northumberland. Guthrum agreed to convert to Christianity and to abandon hostilities against Alfred's kingdom.
A.D. 878. This year about mid-winter, after twelfth-night, the Danish army stole out to Chippenham, and rode over the land of the West-Saxons; where they settled, and drove many of the people over sea; and of the rest the greatest part they rode down, and subdued to their will; -- ALL BUT ALFRED THE KING. He, with a little band, uneasily sought the woods and fastnesses of the moors. And in the winter of this same year the brother of Ingwar and Healfden landed in Wessex, in Devonshire, with three and twenty ships, and there was he slain, and eight hundred men with him, and forty of his army. There also was taken the war- flag, which they called the RAVEN. In the Easter of this year King Alfred with his little force raised a work at Athelney; from which he assailed the army, assisted by that part of Somersetshire which was nighest to it. Then, in the seventh week after Easter, he rode to Brixton by the eastern side of Selwood; and there came out to meet him all the people of Somersersetshire, and Wiltshire, and that part of Hampshire which is on this side of the sea; and they rejoiced to see him. Then within one night he went from this retreat to Hey; and within one night after he proceeded to Heddington; and there fought with all the army, and put them to flight, riding after them as far as the fortress, where he remained a fortnight. Then the army gave him hostages with many oaths, that they would go out of his kingdom. They told him also, that their king would receive baptism. And they acted accordingly; for in the course of three weeks after, King Guthrum, attended by some thirty of the worthiest men that were in the army, came to him at Aller, which is near Athelney, and there the king became his sponsor in baptism; and his crisom-leasing was at Wedmor. He was there twelve nights with the king, who honoured him and his attendants with many presents.
With the long wars over, land was parceled out to Viking soldiers. The Chronicle mentions that the invaders had been turning into "harrowers and ploughers" of the land even prior to this. The land ceded to the Vikings became known as the Danelaw, and there is abundant historical and linguistic evidence that the Viking settlers and the native Anglo-Saxons traded, intermarried, and generally learned how to live with each other.
But it all started with Ragnar and a pit of snakes.
Here's another question: Why did the vikings want to conquer such vast areas of land from England?