How important was the Islamic transmissions of classical texts?

Answer 1

Both critically important and not particularly important at all, depending on the time period and perspective.

In medieval Europe, especially the 11th and 12th centuries and the high point of Scholasticism, most Greek texts reached western Europe via Arabic translations. Greek was almost completely forgotten in the West, and little in the way of literature or scholarship flowed from the still flourishing Byzantine Empire. Instead, what was known of major figures like Aristotle and Galen mostly came to Europe through Islamic Spain, where it was translated from Arabic into Latin. Apart from mere translations, the Islamic world also spurred creative thought, and their own Aristotelian-influenced philosophers (such as Averroes) had a strong and direct influence on medieval western philosophy.

On the other hand, this was only true for a limited time period, and for a limited subset of classical texts. The Islamic scholars who absorbed classical Greek literature were mainly interested in science, medicine, and some philosophical texts; note that I said Aristotle and Galen (a medical writer) were the two major authors to be translated into Latin from Arabic. Very few authors, even among the major names we know today, became known in western Europe via Arabic -- little to no Plato, no Herodotus, no Thucydides, no Plutarch, no Homer, no Sophocles, no Aeschylus, no Euripides. These all became known later, when Greek started to become prominent again (around the 15th century), and they were translated into Latin directly from Greek.