Ancient Greek in the form it was used during the Golden age of Pericles and the Peloponnesian War has been studied to great detail, and we have discovered many extremely specific grammar rules, variations of words based on the region they were used in, detailed and multiple meaning for verbs, and guidelines for syntax. However, paper did not exist during that time, and I think (I might be wrong) that ancient Greeks did not widely use parchment or papyrus to write on. This form of ancient Greek was not spread outside of the Greek peninsula, and another, simpler form of it was widely spread by Alexander the Great 's conquering of Asian regions. As far as I know, it was during that Hellenistic period that it became popular to keep written records of scientific achievements and such, and many of them were stored in the library of Alexandreia, which was burned. It was quite a while afterwards that Christian monks started writing down ancient texts, plays and historical records. Therefore, and assuming that all the information I stated above is indeed correct, how do we have enough written record of a dead language to have been able to analyse it to such a detailed extent?
I think you're making some leaps and assumptions that undercut your premise.
So, first up you're talking about Attic Greek it seems (based on your 'Golden age of Pericles' comment). I think you're wrong on parchment and papyrus use by Ancient Greeks. Papyrus was traded throughout the Mediterranean and would have been one of the materials used for literary production, including works such as Thucydides, Herodotus, etc. - early ancient Greek texts of the type you're discussing. Greek writing becoming a highly literary field and stayed that way.
Secondly, while it's true that Koine was the form of Greek that evolved and spread throughout the Hellenistic period, Koine and Attic are highly mutually intelligible. Moreover, Greek education traditions in the upper classes still had a 'classicising' tendency, and no more was this more apparent than in the 'Second Sophistic', which at minimum was mid 1st century AD through to the the 3rd century, but could be extended beyond that. The 2nd Sophistic was a literary movement which involved increased Atticism - the tendency to revive and imitate Attic styles. This classicising tendency influenced most of the later Greek writers, including church writers, so that Late Antiquity Greek writers are often more 'Attic' than they are Koine.
You seem to overplay the importance of the Library of Alexandria, as opposed to much more widespread copying and distribution of texts. Similarly, while medieval survival of texts depended upon the formation and emergence of European monastic traditions, early Christian groups showed a high degree of interest in 'texts' and were part of Late Antique literary culture too.
There are not, then, such big 'gaps' in the transmission record, and the continual history of Greek, along with its interaction with Latin and other languages, serves to 'underwrite' a general confidence in the ability to read and understand Attic Greek well. As for "extremely specific grammar rules" and the like, some of that is from the continued tradition of Greek studies, some of it has emerged in more recent years with digitisation and corpus linguistics, but it is not particularly vulnerable to a 'gap in the records' as your question seems to aim at.