So the question is massively complicated by the fact that what exactly what a Duchy and terms translated as being a duchy was varied substantially over time and space.

So one important example would be the electors Duchies of the Holy Roman Empire. The amount of power the Holy Roman Empire had over his subjects waxed and waned over history but as a general rule the Holy Roman Empire as a whole was relatively decentralised and particularly the further you got from where ever the de facto political capital at that time was be it Magdeburg, Nuremberg, Vienna, Wetzlar or Regensburg ect. And within the Holy Roman Empire there were often extremely independent making many of the laws within their own territories and retaining a significant degree of power. Later on after the ghost of the Holy Roman Empire was finally exorcised after the Napoleonic wars a number of grand duchies were established which were nominally independent polities although some were more or less stuck under the thumb of the Austrian Empire during Metternich's ascendancy after the Congress of Vienna.

Now compare that to for example English Dukes. An English Duke was typically a close relatively of the king, a son, cousin, uncle. nephew ect. The Duke of Cornwall for example like the Prince of Wales is a title always held by the Heir to the throne since the Black Prince. Generally speaking, with examples like The Anarchy aside England was far more centralised than the Holy Roman Empire with Dukes holding much less power within their lands than a German one might. Over time while the power of the imperial throne generally declined moving into the early modern period over time as England began to set itself on the path to become the bicameral parliamentary monarchy Britain is today power was centralised and the English nobility use their power in the house of lords to exert their influence over the central government of the country and gave up power within their domains to said central government in return for that.

There's also cases like the fully independent "Grand Duchy" of Lithuania which was a fully independent polity during much of the medieval period. I believe there's evidence to suggest the "Dukes" of Lithuania very much did think of themselves as kings and translated their titles as being that of kings. But I don't think i understand the matter fully enough to delve too deeply into that.

Now in terms of the second question. As I hope i've demonstrated what exactly the powers of the nobility were varied between places and time periods. In some places while a noble might manage their land in the property sense royal institutions still played an important part in administering it, judges were appointed on royal authority and the law of the land was relatively uniform. In other places a monarch would have almost no involvement in the laws and administration of the lands of a noble.

Hi!

Short answer? Yes.

Long answer:

We must first understand that differing realms have differing rules, and as such, some kingdoms did have duchies and other titles within their realms. A famous and contemporary example of this is the United Kingdom of Great Britain and Northern Ireland.

As of this writing, there are still 28 dukes in total according to the latest Roll of the Peerage within the United Kingdom, which is the official list of government-recognized peers which both feature Royal Dukes (Edinburgh, Cornwall and Rothesay, Cambridge, Sussex, York Gloucester, Kent) and Hereditary Dukes (all the others)

If we go back much farther, in royal France, we have dukes serving under so many French kings, like the Duke of Alencon who served under King Charles VII le victorieux in the waning days of the Hundred Years’ War or the Dukes of Brittany, who were technically (read: Feudal) subjects of France but are practically a mix of English and local noble control as a result of several wars of succession during the Hundred Years’ War.

Not to mention that there are the more confusing (at least for me) position of dukedoms within the Russian Empire (Grand Dukes vs. Dukes) or the confusing Dukedoms that existed within Greece after the Fourth Crusade.