Except gunpowder, how much more technologically advanced was France/England/HRE etc. in the 12th-13th century than the Roman Empire at it's peak?
With the regression of technology in the Dark Ages, why wasn't information from the Byzantine Empire shared throughout the rest of Europe?
Could a Roman legion of the Julio-Claudian time period have defeated a typical feudal army of the same size without cannons?
Could a Roman legion of the Julio-Claudian time period have defeated a typical feudal army of the same size without cannons?
The stirrup doesn't arrive in Europe until approximately the Eighth Century, and it's very easy to overlook just how much of a game-changer the stirrup was. The key advantage of the stirrup is in the stability it offers a rider, and the ability it gives him to stand up in the saddle. It is basically this which allows for the medieval knight to come into being. The terrifying power of these new armoured cavalry charges can be seen at the battles of Lechfeld and Recknitz in the 950s, where Otto I uses his newly-formed loricati to devastate Magyar and Wendish armies respectively. In your hypothetical situation, a Roman force would likely simply not be prepared for the ferocity of such a charge. If we extend this concept further to the high medieval period when such a charge would likely have been accompanied by massed longbow or crossbow volleys, the Romans are likely to have been severely outmatched.
This doesn't answer your questions directly but there was a large and lengthy discussion on lost technology from the fall of western Roman empire, it's very good, in-depth and well sourced. The commenters of note are /u/grashnak, /u/bitparity who write about the perspective of what was thought to be lost and what actually was lost.
/u/Eursian comments on some reasons knowledge was lost and how some technology was advanced from Roman times.
/U/mrleopards answers a question that may relate to your third question, in that the Roman legions would have met something very similar in Byzantine empire also known as eastern Roman empire with changes emphasize towards armoured calvary.
https://www.reddit.com/r/AskHistorians/comments/838d5s/did_any_medieval_kingdoms_model_their_armies/