For example, in proposition 1, Euclid assumes that the instersection of the two circles exist, when he shouldn't have. This, among many other things, was corrected quite recently (by Hilbert and others, I believe). Were these mistakes only found out then? Did no one mention them for over 2000 years?
Yes. There were numerous commentaries on Euclid in the ancient world, and many of them pointed out flaws in the various propositions. Zeno of Sidon (~100BC) for example, pointed out the Proposition I.1 requires the assumption that unique straight lines can meet in at most one point, which isn't explicitly stated in the postulates or common notions.
Indeed, finding issues with Euclid (and proposing fixes) was something of a cottage industry amongst the neo-platonists of late antiquity.
(as to the specific issue with I.1 you point out, I'm not sure if it was pointed out so early. Heath oddly mentions that it's a "common place objection", but rather uncharacteristically doesn't actually cite any ancient sources that discuss it.)