it's "ὁ ἄργυρος" and "δεδιδαγμένων".
Is "τὰ ἀγαθῶν ἀνδρῶν ἔργα." what the poets were teaching the noble men?
Yes, you got that right.
μένει = remains
You should really use present simple as opposed to present continuous here: it's a general statement. You could even translate as "Neither gold nor silver are forever" (paraphrasing the famous "diamonds are forever").
I'd also translate "γε" as "of course" in this case. But you don't really have to do that.
μενεῖ is future!
EDIT: Oh, and δόξα here means "glory", not "belief".