You could potentially build a large empire with some level of democratic process in place, but not to the extent that you're describing.
The level of bureaucracy you're describing is completely unrealistic in a medieval setting (and that's before you take the problems Cyrus describes into consideration):
- A very small number of people knew how to read or write in those days
- Even many of those who knew how to read or write had no education or culture to speak of. This leaves the door wide open to the manipulation of the masses via clever propaganda, rumors, etc.
- The resources needed to vote on which issues to vote on, and then vote on the issue itself would be absolutely mind boggling. The city would come to a grinding halt as everyone would have to go and vote. Your entire empire would make decisions so slowly that its enemies would be able to crush it without a problem.
- Even today we struggle with ballot stuffing, and dishonesty in the voting system. Ensuring vote validity in a time when there is no good way to discern who's who (no government photo ID) is simply impossible.
Basically .. even having a city run in this fashion is nothing short of a fantasy setting.
So here's how you can - for the most part - accomplish your goals:
A Flavor of Democracy
Your government should work a bit in the same way that democracy works now in the United States (pick and choose some of the elements, of course).
The reason I'm singling them out is because their President is the supreme military commander, and he is able to order certain military actions with no input from any other branch of government. This ability is absolutely crucial in a medieval setting - the ruler must be able to make quick, clear military decisions, and have the armed forces obey unquestioningly.
Some politicians may grumble, the senate may voice their opposition, etc., but they can't stall the decision with pointless political bickering while the nation's enemies advance.
Furthermore, seeing how the US is basically a modern, democratic conquering empire of sorts, the analogy works well.
Uncultured, and uneducated people should not be allowed to vote for obvious reasons (stated above). However, the situation is even more nuanced than that.
Your empire is a bully of sorts. You march into people's nations, bring their governments down, slaughter their armed forces, etc. There's going to be a fair amount of hatred directed at you. Giving newly conquered people the ability to vote is a really bad idea, as you will soon find yourself sitting at the table with elected officials who would like nothing better than to slit your throat given the smallest chance.
So here's the solution: Don't allow just anyone to vote.
Create a tiered citizenship system, a la Starship Troopers. Anyone who serves in the legions for 10+ years becomes a citizen with full voting rights in local and "imperial" (federal) elections. Anyone who serves in the imperial scribe services (political system) for 10+ years can do the same. Many more people may wish to become scribes rather than soldiers risking life and limb, so each applicant should take a test to determine in which branch (s)he would be best suited, and be assigned not only according to their results, but also the need of the Empire.
Those who undertake local vocational training (merchants, cobblers, blacksmiths, etc.) can vote in local elections, but not in federal elections.
Those with no education of any type (minimal wage jobs so to speak) don't get to vote at all.
This might seem harsh, but there's a simple reason for it: the legions and scribe services will serve as educational institutions, and culturally assimilate those who successfully complete their terms of service. This way, the people who eventually end up voting the next supreme leader of the Empire into power do so while fully understanding the issues which the Empire is facing, and who are probably deeply invested in its continued success (since their careers are now tied to its governing body).
Some people are not aware, but most Roman legionnaires were, in fact, foreigners. They signed up for the same reason I describe above: citizenship. The Roman empire would roll in, and like an unstoppable wave, would crush the local tribes. The surviving tribesmen would join up and fill the ranks of the very legions which had crushed their countrymen. Why? Because clearly the newcomers were someone to be reckoned with! They pledged to serve 10 years, which would guarantee them citizenship. This was a huge deal. In the mean time, they became Romanized. They learned the language, they fought the Empire's wars, and gained a sense of pride at the Empire's achievements. You want your own system to work largely the same.
In the end, each conquered nation would slowly embrace your system of governance, although having their populations gain full voting rights might take a few decades.