Origins of Modern English

How did Alfred and other Saxon kings communicate with the Danes? Could they speak each other's language, or did they use interpreters?

Although Old English (Anglo-Saxon) and the Old Norse language of the Vikings were related to each other, they were not sufficiently close to permit anything more than a clumsy pidgin communication. In particular, both languages were inflected, with different case and number endings for nouns and verbs - as in modern German - which differed considerably. The probability, therefore, is that interpreters were used.

Being still largely pagan at this stage, the Vikings did not have ready access to bodies of Latin-speaking clerics, who formed the 'diplomatic corps' of most European states, but both sides would have had persons from a trading background, who were able to at least to converse in the other tongue. One trader, a man named Ottar or Othere, was subsequently received by Alfred at court, entertaining the king with tales of his adventures as far afield as the White Sea.

Following the settlement of the Vikings, a new linguistic chemistry was brought into play. In mixed and border areas, Vikings and Saxons met in the markets or married off their children to one another - and in so doing were forced to communicate. The more closely related words of the two languages, such as  hros / hors - horse, were retained, but stripped of their grammatical niceties.

From this simplified patois of the marketplace, modern English was eventually born.

The English Population

The Celts are believed to have arrived in England around 500BC. The Angles, Saxons and Jutes (southern Germans from Jutland) followed after 1000 years or so with some Viking additions some 200 to 300years later. Apart from the Normans, who were few in number and anyway of Norse descent, there was no mass migration into England until the latter part of the 1950s.
The Welsh and Cornish can claim descent from the Celts, whist the Scots are descended from Celts, who were immigrants from Ireland, displacing the Picts, who inhabited Roman Caledonia. By that reckoning, the English are descendants of at most three races, two of whom are closely related.
The English have shared a common culture for over 1000 years longer than the closest European neighbours.