I was wondering how many Latin (both Classical and Medieval varieties) words have Greek roots. Is Greek the common root of most IE languages?
Not many. The Romans borrowed plenty of Greek words, but mostly in technical senses; in Antiquity, many Greek words that were used in Latin were also considered a bit fancy and special, for better or worse.
There were also some Greek words that were borrowed by the Romans so early in Roman history that they were probably no longer intuitively (or at all) felt to be Greek during the Empire; but the very large majority of Latin words were not from Greek.
In Christian times, some more Greek words were adopted from the New Testament and from ecclesiastical traditions; Christianity emerged in the eastern part of the Roman Empire, where literate people spoke Greek, not Latin (many people were bilingual, though, just as in the western part). But the number of Greek words was still fairly limited; for comparison, the proportion of French or Latin words in modern English is many, many times greater than was that of Greek words in Latin.
Latin is from the Italic branch of languages, along with Oscan and a few more long-dead languages. Greek is from the Greek/Hellenic branch of languages. Both branches are part of the larger Indo-European language family, along with the Germanic branch (English etc.), the Slavic branch (Russian etc.), and several other branches.
While it is true that Latin was influenced by Greek in ways other than vocabulary—such as grammar—, most similarities between Greek words and Latin words you observe are probably due to their common ancestor, the hypothetical Proto-Indo-European language, in which the Indo-European family of languages originated. All Indo-European languages retain much of their Proto-Indo-European heritage.
It is also true that some Indo-European branches are more closely related than others; but I believe the Italic branch was more closely related to the Celtic branch than to the Greek branch. It may be that Italic and Greek are more closely related to each other than to Germanic and Slavic, but they're still not very close.
I was wondering how many of Latin (both Classical and Medieval varieties) words have Greek roots.
As a fraction of words in the dictionary, relatively few, but as a fraction of words in formal texts, not insignificant, depending on the topic and context.
Similar to words of Greek origin in English, which are largely via Romance. Examples would be problem, church, oil, democracy...
Notably, the names of the academic or encyclopædic topics themselves are disproportionately Greek - geology, architecture, economics, politics... As are the words academic, encyclopædic and topic.
But plenty more were just coined in other languages in recent centuries and then only borrowed back into modern Greek - photography, anarchism, Anglicanism...
Is Greek the common root of most IE languages?
No, no living language or subfamily is, they are branches of a tree.
I think that the most prominent word of Greek origin in Latin, and from that into other European languages, is, in its English form, music. The list of other words of course is long, but if I had to choose one that corresponds to a most fundamental concept it would have to be music. There is also cosmos, energy, problem, again in their English forms, and so on, but music just seems such an important concept for humans regardless of culture.