How would administration, educational,the judicial power, taxes etc would work. How would local administration managed when it's inhabitants belong to three separate nations ?
Oh, if one has sufficient support from the powers that be and suitable popular majorities supporting it, I don't see this as any kind of major hurdle at all.
Persons would self-identify as members of this or that group; we can talk about conflicts where lots of persons in a given group don't want to recognize certain persons claiming to be one of them later I guess. Assuming for the moment mere membership in a group is pretty straightforward in most cases and generally resolvable in tricky ones, then each self-identified subgroup would vote for representation regionally and on the Imperial Diet level separately, each assigned proportional shares of the respective body seats by a bureaucracy keeping track. One registers one's group identity with an Imperial bureau with offices in every capital; that tells the bureau how many people of X Y or Z category there are in each bailiwick and thus how many seats of a regional or Imperial body to give them. Everyone votes in common elections, but presenting their respective ID certifying their belonging to this or that group they get a ballot for their group. So in Prauge say, most people are Czechs, or Moravians, but a lot are Germans, some are Jews, some are Hungarians--the Imperial officials overseeing the elections know how many are which by their registrations, so they have the separate ballots ready in appropriate numbers.
In addition to voting for a share of Diet seats, and regional (kingdom/duchy?) and town council and other various boards, of a deliberative/legislative nature, the various defined communities would also select liaisons of some kind to administration. I suppose the Imperial model retains in principle a strong autocratic element; the Emperor is getting "advice" from any deliberative councils, and all executive authority derives from the will of the Emperor, in principle. So if the representatives of any group, or any faction of them, become obnoxious to the Imperial government they can be ignored, censured or silenced. Reasoning along these lines, there might well be a "you cut, I pick" sort of balance in the executive liaisons--the Emperor might put forth handpicked candidates for each group for the group to vote on, or the group constituency puts forth several candidates for the Emperor to choose which is most suitable. I think an Imperial reform demands the former; the Emperor will want to be comfortable with whomever the constituency favors so the Emperor does the nominating.
Looking at say the top level, an Imperial Diet of let us say 300 members, we first apportion the 300 to the various defined ethnic groups, of which there might be say the square root of 300 or say 20; some of these are tiny (Muslim subjects in the lands recently acquired from the Ottomans?) and only get a handful if that; others such as Germans, Magyars or Czechs are numerous and get closer to a hundred. Then one looks at the number of each type in suitable standard geographic divisions of the Empire, and apportions seats. Will the Diet be elected FPTP, or will this reform involve adopting some kind of proportional system? Let's say the reformers dislike the idea of formally recognizing parties but are under pressure to allow for some sort of PR or something resembling it; that would point (among other possibilities) to Single Transferable Vote perhaps. STV was invented by 1900 and practiced I am not sure where; certainly in some American cities by this point, maybe in Australia--it is too early to look to the Irish Republic (known as "Free State" initially) where it was actually the British who imposed STV on the Dail to prevent Protestant/British loyalist minorities from getting steamrollered.
So say out of 300 seats, "Germans" get 90. About let us guess half of these live in the core of Austria, with the other half either in certain scattered countryside settings (Sudetenland) or mostly making up either German-dominated cities in places like Transylvania, or big parts of major mixed-ethnic cities. So Austria itself would get say 50 of these seats (allowing for German minorities in peripheral parts like Tyrolia) and the other 40 are scattered around the Empire, concentrated here and there and with some big catch-all districts for Germans miscellaneously scattered about. So that takes care of 30 percent of the seats. Proceed similarly for the Magyars, about half their seats being in Hungary, the rest more or less scattered, and then the Czechs, Slovenians, Serbs, Jews, and so on. When we get to the smallest group, say Muslim subjects (I should look it all up, this is just conceptual) they might have just two; say one fairly concentrated in the recent conquered zone in the south, and a second catch-all covering the rest of the Empire.
Except we don't want to create a district for each seat; we want groups of seats elected in larger districts. So an ethnicity with just one seat Empirewide will basically elect its single member by Instant Runoff Vote, that being what STV breaks down to when there is only one seat in contention; more typically the seats are grouped in districts electing 2 to 5 members. In principle there are no parties involved, each candidate runs as an independent, but of course in real life everyone pays attention to parties--some independents can indeed be elected. Nothing prevents say the Social Democrats nominating someone to run under their banner for all 300 seats, though it is a reality of STV that a certain amount of strategic management is needed to maximize representation; a party can run too many or too few in a given district and either way fail to get a share of seats proportional to their share of first-choice votes.
Citizens and pundits in the Republic of Ireland pay attention to that metric, unofficially but intensely, much as in the USA we pay attention to total numbers of voters for Democrats or Republicans even though these numbers don't have any legal meaning in the process.
Looking at Irish Republic Dail elections going back to 1982, I observe that the major parties consistently enjoy a 5-6 percent advantage in the number of seats they win versus their share of first-choice popular vote, and small parties are vice versa shortchanged on average, but with a massive chaotic factor meaning a given small party can wind up with twice its PR share or more, or half and less. A premium exists on party unity therefore. Also, gerrymandering is quite possible with STV and IRV, though in this system only an ethnicity so small it only merits one seat in 300 would be facing IRV, and that for a single seat Empire-wide. With just two or three members though, outcomes can diverge pretty far from a proportional assignment of first choices.
One possible workaround for this, assuming the Empire enters a strongly democratic period, would be to recognize parties--or in a variation I personally favor, the candidates form alliances by unanimous mutual consent, and these alliances are the units that are assigned winners proportionally. Sticking to STV as the base, one would add more Diet members, say another 100, and apportion the full 400 by alliance (or party) first choices, then subtract the number of STV seats won from each alliance/party share to get additional members--I would say, instead of a "party list," these be taken from the largest first-choice winners who did not win an STV seat. STV moves the outcomes toward proportionality so this approach is likely to work out pretty neatly.
But of course in the early phase of the reform I expect the powers that be assert STV should be good enough, wherever the dust settles; perhaps instances of outrageous gerrymandering would drive the additional reform later.
Since the statistics of how many subjects there are in each category are driven by individual adults registering their ethnic alignment on their own initiative, there is no need for a periodic census; going into each election, the responsible Imperial ministry would allocate seats to each region for each group according to objective formulas--this might also prevent any kind of deliberate gerrymander; the regions are defined by administrative convenience, and what varies is how many seats each group gets within them. (It will often happen, with the smaller ethnicities, and with major ones in regions where they are less common, that regions that elect say 2-4 members get grouped together in large numbers to elect say 2 of the less common types--say there winds up being just 2 Czech members for all of eastern Hungary, so a portion of Hungary that might be divided into 8 3-member districts for 24 Magyar members elects just those two Czechs.
The Emperor in principle rules on which defined groups are formally recognized, and mediates with their leaders for conditions under which an individual may claim to belong to them. We might have a lot of people of clearly third ethnicity background petitioning to be accepted as German or Magyar. Then again, some ethnicities might lean politically strongly in one or several categories--say Slovaks turn out to mostly vote conservative, but with a strong radical block, but don't strongly support moderate-liberal candidates--certain Slovaks might opt to be counted as German so as to effectively vote for a liberal party they favor, while others might opt Magyar to vote for a fringe ultraconservative party that happens to be strongest if still marginal among Magyars. In practice the Emperor will be guided by prudence and what is politically expedient, but on paper it is entirely Imperial whim that sets the rules for who counts as whom. Again depending on how the system shakes out, at some point some other authority might be agreed upon that can override the Emperor, but if the system works at all well, Emperors will be well advised and tend to keep their prerogatives with a track record of using it wisely. (Much as, in Norway, the King is the final arbiter as to whether electoral regions are reasonably apportioned and whether elections were fairly conducted--if Norwegian monarchs abused this privilege, presumably the Norwegians would either have succumbed to a de facto royal dictatorship, or they would be forced to turn to some other body of umpires, but since the kings have been careful to be fair and reasonable in their judicious use of this authority, everyone accepts their ruling as final and decisive).
So in practical operation, we get an Imperial Diet that technically "advises" the monarch but in practice the Emperor follows this "advice;" in an emergency, or technically any time the Emperor pleases, he can act unilaterally, but this is never done in any normal circumstance; the Emperor practically always affirms whatever the Diet passes. The Emperor handpicks the Chancellor, but again pays attention to the partisan balance within the Diet and looks for someone acceptable to a majority there to pick. The Chancellor has a Council of say 50 persons based on the Emperor negotiating with the self-elected leaders of each bloc of subjects to pick them in fair ethnic proportion; this is the pool from which government ministers are appointed; Chancellor and these top ministers are all theoretically non-partisan and serve at the Emperor's pleasure. But the Government as a whole must work with the freely elected Diet and with another council, 60 or so, of persons elected to "advise" the Government in its executive actions, Empire-wide by the various ethnicities fairly apportioned--say this body has staggered elections, 20 at a time. It isn't a Senate or a House of Lords (there might need to be another body that does represent the nobility, but it need not have a lot of powers, being mainly a pool from which the Emperor chooses Government high officials)--it is the closest thing the Imperial subjects have to a chief executive, and rather far from it. It is a body that the Imperial government might coopt agents from, and which works in theory to harmonize and aid unilaterally decreed Imperial policy, but again the actual ministries can be well advised to seriously consult rather than dictate to this body.
The Diet and the executive Council comprise freely elected representatives of the whole Empire then, but breaks out into separate ethnic groups; the body acting as a whole can vote to delegate particular policy questions to the appropriate affected groups and abide by their caucus resolutions.
Going down the regional ladder, each major subdivision of the Empire has its own levels of councils and administration, the latter remaining creatures of Imperial pleasure all the way down. I would think that cities traditionally had their own autonomously elected mayors, and this would probably still be the case, but in a state of emergency a city might lose this privilege and find an Imperial appointee installed.
Probably it is necessary to continue to keep the monarchy nominally "Dual," ethnicity does not replace this cozy deal between Austrian Germans and Hungarian Magyars, so the Austrian half and the Magyar half of the Empire each have their own Diets.
But come to think of it, I think here is where the "House of Lords" should be--two of them, Austrian and Hungarian. The top level Imperial body is handpicked by the Emperor for a mix of honorific and functional purposes in fair balance from both; those lords so honored who also are assigned jobs by the Emperor form a Lords' Cabinet within the larger Council the honorific lords fill out. At the kingdom level, my guess is that the Austrian body is going to be pretty symbolic and honorific and not much formally involved in the Austrian "duchies," to pick the middle level of honor as symbolic of the whole set (see below); the "duchies" communicate directly with the Imperial Diet and Councils. In Hungary, that is "the Crown of St Stephen" lands, on the other hand, I suppose Magyar conservatism will favor making their Kingdom Lords an active (non elective, obviously, but perhaps with parliamentary electoralism within the House) level, seeking to compel and anyway urging St Stephen "duchies" subordinate and coordinate with this great house, perhaps via a kind of collectiv bicameralism--each "duchy" has a single fused Council but must pass its legislation up to the Hungarian Lords for ratification, and can be directed to act by the Lords. Over time, if the Magyar dominated system proves overbearing or counterproductive, perhaps the Emperor can get the leverage to exempt certain duchies and let them opt for a more Austrian relationship with the Imperial level.
The cities of Vienna and Budapest have special status as Imperial cities, with the city government under an Imperial appointed Prefect; the Imperial family is expected to alternate between the cities. Perhaps over time, a couple other cities--Prague, and perhaps Trieste--can gain the honor of also becoming Imperial Cities, albeit without thereby elevating the Duchy (or Principality or County) they are situated in to Kingdom status. Imperial bureaucracy is spread out between these capitals, though Vienna retains chief place, as the location where the leading Ambassadors of foreign powers are housed and the Ministry of State (or Foreign Affairs or whatever the proper Austro-Hungarian term would be) is located.
At every level where representatives are popularly elected, the principle of apportionment by distribution of registered ethnic group holds; if a particular county happens to be half Muslim, half the representatives elected within this bailiwick will be Muslim. The kingdom level is an exception; there is no popular election at that level and instead it is the center of hereditary (largely) nobility, which would tend to be German and Magyar, with any lords of other ethnicity probably veering toward one or the other of these cultural identities in practice.
Each kingdom is then subdivided into tracts variously called "principalities," "duchies," or "counties" depending on size, wealth and historic dignity, and the councils of each, fusing the executive and legislative/deliberative functions are serious working bodies, with little reference to the formal kingdom Diets. Within these, they are subdivided into small administrative units which are grouped into larger or smaller provincial assemblies; the elected councils here have no legislative function, but are small executive liaisons with the Imperial appointed bureaucrats who have executive authority.