So, there are layers to this.

Yes there are quite a lot of primary documents from antiquity. The easiest ones to find: engraved texts. That's how we managed to recreate for example the Res Gestae, a kind of autobiography/propaganda made by the emperor Augustus: we found several copies of it in ruins.

Then, texts on paper: we have those too. Quite a lot actually, but not from the region of Rome or Greece... but Egypt (which was under Greek then Roman control). The dry climate is excellent for the conservation of papyri. So we regularly find new fragments there. Often it's everyday stuff (personal or even legal writings like a will) which are a gold mine for historians but there are also quite a lot of fragment of ancient texts. Which help us with the next part: how can we know if our texts are exact?

So. First thing to understand: with very very few exception: we do not have complete primary documents from antiquity. We rely on copy and sometimes fragments that concur with the copies. Which for example help us establish that one major text from Antiquity had a fixed form relatively early in its history: the Bible. It also confirmed that our Homeric texts are also correct (since those were used in school we have quite a lot of them). And no, ancient text did not survive a thousand years, they were regularly copied.

So with taking that into account how can we know that we have the exact copy? We usually can't and we need to accept that. Usually not because of malicious intent, but due to mistakes. What we can have are usually close approximations though. First, we compare texts. We trace the origin of manuscripts and see if they have the same text: if a manuscript copied from a French manuscript and one copied from one that originally comes from Constantinople what's more likely is that it's the correct one. Same idea with differences, if only one text differ we assume that it's wrong. If one version is more complex we assume that it's the right one though (since simplifying is more likely to happen that increased complexity). And so on and so on and so on.

Now censorship. To be honest it had happened at times, it's pretty blatant (the usual non christian writing that says stuff like "and in Judea came a man that was the true son of the One God" and stuff). But usually it's rare and we see it. So don't worry about that.

I want to add one final thing though, that underlines your question... Christian monks and Arab copyists are not in anyway more likely suspects than your average ancient copyist. If they did not like a text, or did not care, they did not record it (this actually happened before the Christian era too, books needed to be copied regularly, it's long and expensive). If they copied a text, that's because they had an interest in it. And medieval scientist and philosophers would not dare modify ancient texts, why would they? Those are their work tools. They know they were pagans, and may try a lot in their commentary to justify that "no if they knew about Jesus/the Prophet they would totally have converted" or so. But they didn't touch the text (again the exceptions we have are far and few).

There are for example, the Herculaneum Papyri (buried by the eruption of the Vesuvius, but which xray technology now allows us to decipher). Egypt, too, has delivered many documents preserved by the arid conditions, and of course, we have a score of inscriptions on hard surfaces like stone.

As to the veracity, consider this article , it does a nice job explaining.