I read once that European medieval rulers would use the collapse of Rome as a justification that democracy/republicanism didn't work, thereby further legitimizing their autocratic rule. Is there any truth to this? How much might the average citizenry and the nobility know about Roman or Greek democracy, or rule by an elected body?
Not particularly. First of all, if by the "collapse of Rome" you mean the decline of the Western Roman Empire, it wasn't particularly democratic or republican, and the inklings of feudal structure were already there. Roman Empire was also hereditary (in theory).
Second, the "rule by election" did not disappear in middle ages - people still voted, sometimes for local officials, sometimes for the guildmasters.