Yeah, the first question could be a hefty tome.
But since there are so few Medieval or archaeological topics, I'd love to jump in on this one.
First, the reason why William felt compelled to replace the Saxon aristocracy was because he had so many promises to fulfill. Even after Edward's death, it was still considered crazy to invade A-S England, especially for a Duke, so William made a lot of promises and did a lot of wheeling and dealing to build an army. No one would have attempted it with Edward as King. Now, not only was the battle hell, it was fought and won on October 14, but William was not crowned until December 25. He almost died himself from general illness during the intervening time, there was malnutrition, dysentery, and almost certainly food poisoning. What did they do? Roamed the countryside staying out of the woods, at least for a short while. William secured several towns along the way before arriving in London. So there were a lot of prizes to meter out after going through all that.
As to the battle, William got a number of big breaks that allowed him to win. I estimate he wins that against Harold II Godwineson maybe two in ten times; that was one of the two. The Saxons were not having a hard time repelling the Normans from the top of Senlac Hill (from "sangue lac" = blood lake) until a wing of Godwineson's army got impatient and pursued the Normans off the hill where, on level ground, the Saxons were cut down. That was a major blow that thinned out the shield wall. Plus, Edwin and Morcar were still to arrive from the North. After all that, William still needed all day and one final push up the hill with everyone he had left at dusk to break through the shield wall.
What were the impacts? Language and culture. Anglo-Saxon (Old English) was basically Old German. So, today English would look a lot more like German than it does now had Harold held - probably with a dose of Latin, making for what to us would be a quite strange language. And if there are alternate universes, quite a few of them have this. England likely would have trended more in the direction of Scandinavian culture, since there was and actually still is a great amount of Scandinavian and Germanic DNA in England. Minus the French influence for centuries, the culture would be more Scandinavian. So no Cor de Lion? Nope. No Tudors? No War of the Roses? No Shakespeare? Likely not.