I can't answer all of your questions, but I hope this answers a few

William the Conquerer was part of a struggle for the claim to the English throne in 1066. In the year 1066, Edward the Confessor, who had been king, passed away. This would not be a problem, except he had no heir, as he had a vow of piety. As such, this led to a battle to the for the throne.

Three people emerged as possible heirs. First was a Norwegian king, Harald Hardrada. Hardrada had a claim to the English throne from an agreement between a prior Norwegian and Danish king, Magnus the Good, and a former English king, Harthacnut (son of Cnut the Great). This agreement stated that should one of them die, the other would take over the throne of the country that the first ruled over. Hardrada believed that this carried over to his rule as well. Not that it trully mattered either way; Hardrada was willing to take England by brute force as was.

Second was an Anglo-Saxon, Harold Godwinson. Godwinson was the only one of the possible heirs actually living in England at the time. His claim to the throne was that he was Edward the Confessor's brother-in-law, with Edward having married his sister, Edith. Given that he was the only of the heirs in England, Godwinson was named the king immediately following Edward the Confessor's death.

Finally, we have the one you're looking at, William the Conquerer (or William the Bastard, whichever you prefer). William was a cousin of Edward the Confessor and was linked to Emma, who had been wife of the English king Cnut the Great. This gave William, too, a shot a the throne.

After Godwinson was crowned king, William began plotting his own attempt at the crown. Here is where some luck comes into it. At the same time, Harald Hardrada launched his own quest for the English crown. Hardrada sailed from Norway to Northumbria, where he began advancing toward the city of York. Godwinson rushed his troops north to face off with the Viking invaders, leading to the Battle of Stamford Bridge. Godwinson would win, and Hardrada would die.

While all this was going on, William officially began his conquest, setting sail from Normandy. This, in turn, forced Godwinson and his Anglo-Saxon army to have to immediately turn around and head south. This leads to the Battle of Hastings, where a tired Anglo-Saxon English army was forced to fight a fairly fresh Norman army. The Normans, of course, won, Godwinson was killed, and William took his place as king of England and effectively ending the age of the Anglo-Saxon.

As to how William was able to keep the king of France out of this: Normandy had been somewhat independent from France thanks to the Vikings. During the 9th century, numerous Viking raids occurred in France. To get these raids to stop, several French kings gave tribute to the Vikings, eventually culminating in King Charles the Simple giving a Viking leader, Rollo, the land that would become Normandy.

My main source for this is A History of the Vikings by Gwyn Jones.

If others notice something I said that's wrong, feel free to chime in.