A gripe I've often had with internet historians is the idea that "Everyone always knew xyz, because several Greek scientists were able to deduce it.", while the latter sentiment may be true, this doesn't mean that society as a whole subscribed to this belief, or if they ever heard it. How well did information from books, poems and other philosophical works disseminate?
Did the Greeks all know Herodotus' writings on Egypt? Did the Romans share Ovid's Ars Amoria with each other like it was a Cosmopolitan article? Were Pliny the Elder's facts on nature common knowledge?
What kind of numbers was Ciero getting?
*Edited to remove a misleading point
Estimating readership of Latin texts is very difficult: papyrus (on which the texts were mostly written) degrades, unless it is left in a very dry environment. So apart from localised areas of extreme heat and dryness (such as the aftermath of Vesuvius), papyri did not survive in Rome and Greece. Egypt, however, has permanently had a nice arid climate (south of the Delta at least), and this means that lots of papyri survive from there: mostly administrative documents, but a large number of literary texts as well. Unfortunately for three of your four questions, the Romans never inhabited Egypt in large numbers, so the number of literary Latin papyri we’ve found is tiny, but does confirm what we would suspect anyway: the Aeneid is the best attested Latin text.
For your first question (about Herodotus) we’re on firmer ground, because we have thousands of Greek literary papyri, and if we count the number of examples for each author, we can get a fairly good grasp of how popular different writers were (at least for Greeks in Egypt during the Roman Empire, when most of our papyri are from). Homer is the massive outlier, with nearly 1500 specimens (data taken from Mertens-Pack’s database of Greek and Roman Literary Papyri); of these (largely contrary to modern taste), 1200 or so are from the Iliad, and only around 200 from the Odyssey. Homer was omnipresent in Greek life, and was the first Greek text you would be taught to read (ideally); Hunter notes that gravestones from the middle of Anatolia in the fourth and fifth centuries AD show that even these very rural, very Christianised farmers were still reading and trying (very badly) to imitate Homer. So, given the omnipresence of Homer, the figures for other authors’ papyri need to be placed in context.
The next most popular poet is Euripides (this is much more similar to modern opinions), particularly his Trojan Women (anything Troy-related tends to have gotten relatively big numbers), with about 130 papyri; Sophocles and Aeschylus have about 30 each, Sappho 20, Pindar 50, Hesiod 110 – no-one comes remotely close to Homer, and Euripides stands out quite a bit as well.
No prose author gets anywhere close to Homer either, but they do get similar numbers to the major poets. Herodotus actually seems to have not been really popular (which, as West notes, has been a surprise to modern historians since they started digging up papyri) – about 40 examples, so more than any playwright other than Euripides and Menander, but less than Pindar and Hesiod. Of the other major Classical historians, Xenophon also has around 40 (though he wrote a lot more than Herodotus), but Thucydides has 90, which is a big number, and indicates that he was (relatively) well read. Plato also has around 90 (again, as you might expect), but the two best attested prose authors are not well known today: Isocrates (no relation to Socrates) is just under 100, while Demosthenes goes even over Euripides, with about 165 papyri.
All of this brings us back to Homer, and the purpose of these papyri. Homer might have been the first, and most important, author you would read as part of your education (for a variety of reasons), but he wasn’t the only one. During the Roman Empire, Greek culture started to put a renewed focus on the art of rhetoric, and particularly rhetoric from Athens during the 5th-4th centuries BC (this cultural movement/period is called the Second Sophistic). Because of this, when you were growing up you had to learn how to read, write, and speak texts from this period and of this kind, and so to make it in life you had to read, and reread, the relevant authors, primarily Aeschines, Isocrates, and Demosthenes. Thucydides was an Athenian but, more importantly, wrote in the Attic dialect (unlike Herodotus, who wrote Ionic), and wrote very specifically about Athens towards the end of the 5th century BC, which made him very attractive as an educational model for multiple reasons (in contrast to the much less focused Herodotus).
So to answer your questions: non-fiction was very widely read in antiquity, primarily works of history, oratory, and philosophy; certainly not all Greeks knew what Herodotus wrote about Egypt (and to some extent, after they had immigrated there themselves they didn’t need to know), but he was a relatively popular author.
Secondary Sources
Houston, G. W. (2014), Inside Roman Libraries: Book Collections and their Management in Antiquity (Chapel Hill)
Hunter, R. L. (2018), The Measure of Homer: The Ancient Reception of the Iliad and the Odyssey (Cambridge)
Mertens, P. & Strauss, J. A. (1992), ‘Les papyrus d’Hérodote’, Annali della Scuola Normale Superiore di Pisa 22: 969-78
West, S. R. (2011), ‘The Papyri of Herodotus’, in Obbink, D. & Rutherford, R. (eds.), Culture in Pieces: Essays on Ancient Texts in Honour of Peter Parsons (Oxford): 69-83
Not sure if this is exactly what you're looking for, but knowledge about things like this did percolate to common people -- this older answer addresses the flat earth question. Obviously educated people who would read these texts in their original form were a distinct minority of the population, but ideas about how the world worked percolated throughout society.