How far back did the English have an accurate understanding of the history of their own people?

You'd need to define the term accurate here. Even today depending on how you define it the answer is going to be somewhere between 100BC and 8th/9th century AD.

Did they know their ancestors came from what is now called "Denmark"?

Mentioned by Bede and a lot of the later histories derive from him so yes-ish. Its not clear how exact they were in terms of location.

When did accurate historical knowledge turn into religious myth for 10th through 14th century English people?

Err 1914 Angels of Mons? The earliest history that there would have been somewhat accurate knowledge of was Caesar's invasions of Britain in 55 and 54BC. However even today figuring out what what was going on between Caesar's invasions and the Claudian invasion of 43AD is rather difficult.

King Arthur would have been in the standard histories but so would tales about the later 9th century Edmund the Martyr (actual solid history limited to he was king of East Anglia when the Great Heathen Army conquered it and he died as a result).

So anything before 55BC is myth (Largely from Historia Brittonum via Geoffrey of Monmouth's Historia Regum Britanniae). After that you get a mix of history and myth although Historia Regum Britanniae leans heavily towards myth and it had a lot of influence on later histories.

14th century histories still exist Ranulf Higden's Polychronicon for example. The scottish Declaration of Arbroath is also 14th century and that contains a wonderfully mythical bit of history as to how the Scots came from Greater Scythia before genociding the Picts.