That is, maybe at Herculaneum or Pompeii, more modern archaeology uncovers something like "first editions" of the works of the more famous Greek and Roman philosophers or historians, and it's been found that the transcribed versions that became famous again in the Renaissance were not the same.
Not "first editions" (haha), but ancient copies, certainly. The most common avenue for finding ancient copies is via papyri preserved in Egyptian rubbish dumps, or turned into papyrus-mâché and used in sarcophaguses, though papyri have certainly turned up elsewhere too (e.g. in Herculaneum, as you mention).
They're nearly always very fragmentary. Often the surviving fragment preserves no more than a few letters. But yes, they're important sources of textual variants. They're not always better than the mediaeval manuscript tradition: transmission errors happened in antiquity too. But earlier witnesses always add more information, so even if there are errors, they're always informative.
The number of papyri that have been found varies depending on how popular the text was. For things like the New Testament and Homer, we have thousands of papyrus fragments. For example, we have fragments of at least 136 separate papyri of Euripides, 36 of Sophocles, and 32 of Aeschylus. For early epic, we have 1569 papyri of the Iliad, at least 252 of the Odyssey, 58 of the Catalogue of Women, 40 of the Theogony, and 32 of the Works and Days.
A very few are relatively intact. There's one 5th/6th cent. CE papyrus that preserves substantial chunks from every book of the Iliad, and one 3rd/4th cent. CE papyrus that preserves most of 11 books of the Odyssey. A very few texts are known only from papyri: the main ones I know of are Aristotle's Constitution of the Athenians, the majority of what survives of the Catalogue of Women, and the Derveni papyrus.
As to how they shed light on surviving texts, I wrote an old post here which gives one example of how a papyrus reading affected our understanding of the correct reading of a text. Papyri are and will always continue to be an important avenue for finding new corrections to ancient texts, which is why there are a few journals specifically devoted to papyrology. In a modern critical edition of a text, papyrus readings will normally be annotated in the apparatus (the list of textual variants at the bottom of the page) with the letter p, though the exact typography varies from edition to edition (e.g. sometimes an edition will use a lower-case italic p; some older editions use a capital P in blackletter script).