Firstly, there is important distinction to be made between roughly two broad categories of cities, (1) those with relatively stable ( Roman ) continuity and others communal cities, to those from (2) high to late Middle Ages, usually established as imperial cities or by other territorial lords, although this differentiation can be somewhat problematic if generalized too much.

So, their respective governments and various privileges will differ accordingly. If we take a look at northern Italy and the cost of Istria, and compare these cities with continuity to those established later, and how the legal and administrative developments changed after their adaptions to continental "feudal economics", what we see when these cities fall from Langobardian, and later Byzantine, to Frankian and later Holy Roman, and lastly to Venetian, speaking about Adriatic coast.

So, a rough example from both "types" of cities;

(1)Venice

Venetian assembly, who was firstly comprissed of free citizens ( arengo ), elect a Doge ( duxe ), but this free assembly already loses the power by 1130s, when the election would go through 40 electors, who were appointed by an 11 commissioners.

The main body was Maggior Consiglio, which was firstly implictly, but by late 1290s explicitly, limited to city nobility, but due to the size and impracticality of such a large entity, by the Doge´s appointment, thus called Pregadi, a smaller, firstly of 60, later 120, members to offer more flexibility and responsiveness - sometimes called a Senate.

The highest organ was the Council of Ten, -- the Doge, and the six advisors appointed to him, together with elected Saggi from the senate, comprise the Minor Consiglio. Lastly, the Doge and his six advisors, together with a body of three ( from an organ of Maggior Consiglio ) would further comprise Signoria. These functions were largely hereditary, of Patricians, who usually also had agrarian property, so even simply being a citizen of a city would not give you much governmental options and influence, since participation in them was strictly limited.

But the tradition of other communal cities, which would by the roughly 13th century come either under other City republics, like Venice, or others, like Trieste under Habsburgs, would go through slight modifications, namely, there was usually an apointee, the Venetians called them Potestas et Capetaneus, Habsburgs Hauptmann, but their role was largely limited to millitary function, and civil matters were left to the cities themselves, although between them, Venetain were definitely more intrusive due to the the geographical positions.

The Communal cities had their own Statutes, so their respective organizations were described there, but any generalizations here would be misleading, since they differed substantially and went through significant changes and developments compared to Venetian system. I can say more about an example of this if requested for some other Adriatic cities.

(2) Continental cities

On the other hand, continental cities were established by an act which gave certain privileges to existant settlements, markets, communitas civium or Gemein der Burger etc. where initially, territorial lord would have the main function, although already here, a notable difference has to be made between cities under a terrotiorial lord who was likewise a Provincial lord, in the case of a duchy, a duke, and other landed nobility and cities established or acquired by them, which influenced their developments, since other landed nobility did not necessarily have the provincial jurisdiction. The city was run by an appointed iudex, who beside the city, had larger provincial jurisdiction in certain matters.

The general assembly of citizen was Banntaiding, which had initially both legislative function and other common present issues - a somewhat judiciary function - although these seperations would not make much sense at the time, so this is rather anachronistic - usually, though not necessarily, under the appointed iudex, typically 1-3 times per year. From this main body, a smaller set of 12 would be chosen to form Geschworene, which would be in session far more regulary under the iudex - limited to the upper social spheres of the cities - though not hereditary or otherwise limited as seen in Venetian example. The development of city Patriciate was limited in these continental cities ( unlike Venice and other such cities ) due to the lack of agrarian land, and until 14th century it seems there was such a possibility, but the Habsburgs quite strictly limited this development. During this period, city goverment slightly changes, both the city judge ( iudex ) and Geschworene ( Notranji svet ) grows more independent from the terriotorial/provincial ( not synonymous ) lord, and these two organs serve as a barrier towards other wider city organs, and tread the line between the lord and other city organs, which in some placed did manage to get their representatives to witness the session of Geschworene, although they did not actively participate - and the largest city assembly by the late 1400s falls appart to smaller organ - limited citizen assembly. The judges are elected by recommendations of two candidates ( one of them would typically be the sitting judge ) by the of Geschworene by other two organs ( representatives and limited assembly ). Other city offices ( like notary, fiscal, etc ) were mostly done by the members of Geschworene, or persons appointed by them. The development of these offices would greatly vary city by city ( location, population, fiscal capability, etc ), mostly formed from 16th century onwards.

We can expand on this, although I am more familar with Northern Adriatic cities and Habsburg lands. And to adress the subquestions;

How did the nobility react to having to share power and status with lowborn leaders?

This formulation is certainly somewhat problematic, cities as entities certainly were not "leaders", but that is not to say there wasn´t any friction present, landed nobility tried to influence city trading and control their own tenant population and their interactions with them, but on the other hand, cities were lucrative, and contibuted a substantial sum of provincial taxes, so a consensus was tried to be reached.

The competition between the cities themselves could be heated, trying to get exclusive permissions of trade to some materials and luxuries. For example, Vienna in 1366 exluded Laichbach from Venetian trade, or how a year later Pettau was exluded from Hungarian trade, which needed to travel to Vienna - Vennice. Such disputes were relatively common, and sometimes required imperial intervention to be resolved - or arbitration.

Would their institutions be recognisable to us as "democracy"?

No, not in any nowadays sense of the word of what we typically mean by it, nevertheless they had successive organs by raising authority, but they were conditioned in such a manner that excluded vast majority of citizens.