I have long been fascinated by the idea of lost historical texts, and mourn the loss of vast swathes of works of famous philosophers, historians and statesmen (including sometime whole libraries from antiquity). Famously, many of surviving Latin and Greek texts are themselves translations of translations of copies that were saved from being lost forever from sources outside Europe, such as those preserved by Arab scholars. What chance is there these days of major texts, such as a work by Plato or Cicero, being rediscovered? Does it ever happen? If so, where are such things found? What is the most startling such discovery of our time?
Only a handful of extremely significant Greco-Roman sources have been "uncovered" since the Renaissance, and I use "uncovered" with discomfort, since the vast majority of Greek sources in particular never had to be "uncovered" except in the sense of being made known to people who hadn't previously known them. Some genuine discoveries have been
These are ones that are particularly important for me; others would no doubt have their own choices. A few other favourites might include
Extremely significant discoveries of Latin texts are much rarer. A dozen or so lines of Gallus' poetry hardly qualify. Certainly there are a few works that had only a very, very tiny circulation until modernity, like Catullus' poems; and there's Claudius' speech at Lyon, which was completely lost and rediscovered in the 1500s. These aren't exactly current events, though.
Now, all that said, I want to clear up a misconception in your question:
many of surviving Latin and Greek texts are themselves translations of translations of copies that were saved from being lost forever from sources outside Europe, such as those preserved by Arab scholars.
This is almost entirely untrue. Arabic translations of Greek texts (not generally Latin ones) were in the Middle Ages an important vehicle by which mediaeval Western Europeans obtained knowledge of these texts. But no modern editor, and no mediaeval scholar in the Byzantine world, would dream of assigning much importance to those versions. That is because the originals never went out of circulation in the Byzantine world. For most students of Greek antiquity, Arabic versions aren't even a blip on the radar -- except in a few genre-specific cases and for some other very, very specialised purposes.
I know this won't answer the majority of your questions but I'm currently doing my thesis on L. Licinius Lucullus, the Roman who led the charge against Mithridates VI Eupator of Pontus in the first century BCE before being replaced by Pompey, and Plutarch wrote that he wrote a history of Rome in Greek during his lifetime. Plutarch may have used this as a source for some of the details in his biography of Lucullus but since it hasn't survived it's impossible to tell what's true and what Plutarch was inventing for his own reasons. That said, in during the Roman Republic it wasn't uncommon for Philhellenes (the members of Roman society who appreciated the language, culture, and art of the Greeks) to write things in Greek that to this day haven't survived.
As far as I know there have been a significant number that haven't survived and it would be really quite amazing for any of them to pop up out of some archive somewhere, let alone as some form of primary source. In my limited knowledge it's rather uncommon and what has been found lately is from much older sources and as far as I know the Oxyrhyncus papyri is among the most significant finds of recent times. They include various fragments from ancient poets, dramatists, and even ancient Biblical manuscripts that help fill out the catalogue of ancient source material.
Needless to say as far as I know the chances of coming across something entirely new is extremely slim.