Some added thoughts from here:
Technologies & Policies:
They were the same thing in Civ VI, now technical/scientific advance and advance of society are somewhat seperated. I think Civ V itself does not stick to this seperation quite well. E.g. Chivalry is in my opinion not a tech, but would fit very nice as a Policy branch.
From what I understand, the most striking trends in Medieval ages were Feudalism and Piety and everything that goes with them (serfdom, manorialism, chivalry for the first, crusades, divine right, monasteries, sacral buildings for the second) and I think it is these things that we should cover with the Policy system rather than the Tech Tree. (Currently, the Policy system lacks relevance apart from the cultural victory, which is a pity). Also striking is the division of people into 3 classes: Clerus, Nobility and the common folks.
We could redesign the policy trees to give the player the choice to support one of these groups with choosing a tree, Nobility (including Knights) boosting military and "land" economy, Clerus boosting happiness, culture and science, and the common folks boosting "city" economy and science (we're talking about the founding era of most modern cities after all), maybe only available from high-medieval era.
so, initial proposals for some policy branches:
Serfdom - Increases Farm output, Chivalry - gives nasty promotions, Vassallage - reduces army upkeep
divine right - Increases

, Sacral Architecture - increases wonder

, Scholarship -

for monk/priest specialist (merge artists and scientists?)
Guilds - extra

/

for engineers / artists, City-alliances - increased trade route yield
and so on.
Victory Conditions:
Domination Victory can be kept as it is obviously, but the others must recalibrated. I like the Idea by rf900
but I think it could be broadened a bit, the medieval ages did not end with one event, rather one could have the player build Christopher Columbus Expidition, Gutenbergs Printing Press, and some gunpowder thing and combine them to win.
As for the cultural victory, It would propably be OK to remove it from the game, but we have to give the social policies enough oomph to be tempting still. Or we'll leave it as it is and call the Utopia-Project "Reformation"
Economy - Cities and Country:
More than ever, people live in the countryside rather than in cities in the middle ages. Civ is propably the wrong game to represent this in any way, yet one could at least consider reflecting this with increased focus on improvements. Villages, Monasteries and Castles come to my mind immediatly. Can acces to these be tied to policies rather than technologies? I don't know.
Buildings and Units:
Putmalk did a good start there. I don't understand how you could miss the spearman though! I don't see any fault in going for Spearman / Axeman / Swordsman and maybe heavy versions of these, if gameplay permits. I don't think the Peltast fits the era, maybe we'd be better of with two versions of archer and crossbowmen.
As for buildings, many of them are unfitting in their name, but necessary in their function. Some suggestions:
Monument - Chapel, Temple - Church, Opera House - Cathedral (I guess one could come up with something less boring)
Colisseum - Tavern, Theatre - Tourney-Place, Stadium - Theatre
Library and University are ok, as are Market and Bank.
Okay, enough for tonight, I hope you liked reading!