Yes this is the passive voice. Active: The English submitted to William. Passive: William was submitted to by the English. I don’t think this was written tongue in cheek. Here is a rewording: The English who had lacked leaders (and had been conquered and had people taking the throne by force so much recently) soon submitted to William the Conqueror (The pope supported William’s claim to the throne) History is one of my favourite subjects, Let me explain why we submitted to the conquerors. Between Alfred the great (about year 800) and William the Conqueror (1066) we had many rulers. The Danish and Vikings had raided and ruled the eastern part of England for quite a while, but there was fighting all the time. And the Kings who eventually ruled were all very weak. William had a claim to the throne because he was a cousin (I think) of the king (Edward the Confessor, very popular king) and had lived in England for a long time. Also King Edward named William as the next king. We find out that by 1086 (Doomsday book) everything had settled down. Sorry for the long answer, I hope it was interesting.

Well, you begin with William the Conqueror as the topic of the sentence. However, all the actions refer to what happened to him, not what he did. So you use passive voice (be + past participle) to express this. Halfway through the writer describes what happened to the English. Again, passive voice is needed.

Yes, the main clause is written in passive voice. This is history written tongue-in-cheek. In other words, it is not intended seriously. I don't think the English at Hastings welcomed William.

You are right and this site gives a clear explanation of passive and active voice http://www.englishclub.com/grammar/verbs-voice.htm