- When did the Mycenaeans overthrow the Minoans? ca. 1400 BC.
- When did the Minoans disappear eventually? The last archaeological site displaying Minoan cultural features indicates 1000 BC. The Eteocretan language, which may be a continuation of Minoan (by default, I’d assume it’s likely), survived down to the third century BC.
- What was the cultural role of the Minoans since? When the Mycenaeans subjugated the Minoans, they were a subjugated underclass, which preserved at least some of its culture. The memory of Minoan overlordship of Greece seems to have been preserved in the myth of the labyrinth (a pre-Hellenic word), which involved Athens paying tribute to Cnossos.
Doubtless there is a considerable historical element in the legend, perhaps in the Phoenician origin of Europa; it is possible that not only Athens, but Mycenae itself, were once culturally bound to the kings of Knossos, as Minoan objects appear at Mycenaean sites. 
[After the 1450 BC eruption of Santorini] The palace in Knossos seems to have remained largely intact, resulting in its dynasty’s ability to spread its influence over large parts of Crete until it was overrun by the Mycenaean Greeks.
Minoan palace sites were occupied by the Mycenaeans (who adapted Linear A Minoan script for their own language) around 1420–1375 BC. Their language, a form of Greek, was written in Linear B. The Mycenaeans tended to adapt (rather than supplant) Minoan culture, religion and art,continuing the Minoan economic system and bureaucracy.
When the warlike mixed group conventionally referred to as Dorians arrived on Crete from the Peloponnese after ca 1100 BCE, archaeological reconstructions suggest that they would have found the Minoan people living along with the Mycenaeans, surviving as an underclass. No doubt the Minoan language continued to be spoken by the peasants, though inscriptions, now in Linear B, were all in a form of Greek associated with a Mycenaean upper class (BBC). The Dorians seem to have driven the local people up into the hills; the latest towns with Minoan material culture are in more and more inaccessible places, one of the largest and most extensive settlements being at Karfi, high in the Dikti Mountains. At this high, remote, ancient, and sacred site a fragment of Minoan civilization survived intact for about 400 years after the occupation of Knossos.
That takes us up to 1000 BC.
In eastern Crete about half a dozen inscriptions have been found which, though written in Greek alphabets, are clearly not Greek. These inscriptions date from the late 7th or early 6th century down to the 3rd century BC. The language, which is not understood, is probably a survival of a language spoken on Crete before the arrival of Greeks and may or may not be derived from the Minoan language preserved in the Linear A inscriptions of a millennium earlier. Since that language remains untranslated, it is not certain that Eteocretan and Minoan are related.