For me, the Duke is responsible for the security of his area of operations, as well as insuring that the worlds toe the line of Imperial desire, and that no one world gets out of line with another to the point where its independent planetary governments start going at it hammer and tong because of issues that would have led to war absent the Imperium's existence.
I agree and disagree with your point. It was made clear in TAS News Feeds by GDW that Nobility were intrisically involved in massaging Flash points and potential combatants to working together. This does not require or even need Naval support so I disagree with this being joined to a discussion of fleet ownership except at the extreme levels. A CT Canon example of the extreme level would be the Porozlo(Rhylanor) System and their use of a planetary navy to prey on the merchant fleets of neighboring systems. As a result, the Imperium stepped in and squashed fleet and placed on that system a limit of being unable to own warships. Here the situation is so exteme that non-IN warships would be needed long term and would be squadrons of ships owned/controlled by local Dukes, Counts and possibly effected Marquees. So the Porozlo Squadrons(represented in the FFW boardgame) would have major capital ships owned by Duke Rhylanor(note, this is a Heraldic Title, not an attempt to mis-name Leonard, Duke of Rhylanor) supported by lessor ships from the personal navies of Count Jae Tellona and the planetary navies(if they exist IYTU) of the Systems Belizo, Gileden, Fulacin(Vacent: 1107) and Tureded. IMTU, these ships are what considered the "Colonial" fleet in the region(also as supported in the board game Fifth Frontier War). They would, as well as adding to a Nobile's "bargoning strength" enhance and fill in for the IN. Like the Coast Guard and the Naval Reserves in the US Armed Forces, these lesser/older ships would handle lower priority or more local missions with locally harbored resources. They also fill in the longer term missions such as "drift anti-pirate patrols" and the like.
GRAND FLEET makes sense to me in a large respect primarily because of the material in HIGH GUARD. Truth is however, that not all Duchies are created Equal, and as such, it seems to me that the buget constraints of one Duchy might not be the same as those found in a more affluent duchy elsewhere.
This is clearly in Canon with examples such the Glisten Duchy, who's Duchess is tasked with overseeing the improvment of conditions in Dist. 268. So a major slice of the economic pie for Imp activities would accrue to that Duchess/Duchey with permission, From the Emperor himself, to expand her military to support Client States in the area(Collace, Tarsus, etc...), face down opponents(Trexalon) or maintain expected future boarders with growing alien polities(The Mewey). The Duchy of Glisten has florished under an increased role as Guardian of Dist. 268 where the Duchies Lanth and Lunion see more of their money funnelled directly to regional Imp Navy fleets. This may change if the Emperor designates one or the other Duke as admin for the Border Worlds. Such a decision would route economic traffic through the selected Duke's Court. To better see this, I recommend reading the histories of the various courts and gentileman criminals, of Europe.
It also seems to me, that in an effort to appeal to the self-interest of the man who administers to the Duchy, that he be given power not only on a political level, or miltiary level, but also at the administrative and/or economic level.
Interesting but short on one level. The use of power need not be even in all types. An Imp Governor needs diplomatic might but no military power at all. They back by the might of the Imp Navy. Said Governor does not have power over the Navy but can call on them as needed and answer to the Emperor afterwards to justify it. So the Governor is a Diplomatic Power in the area with administartive power to handle oversight of Imperial laws placed on the World Government(a 100% seperate gov't entity...remember, The Imperium owns the space, not the planet.). So the member government deals with the Imperial Governor regarding Imperial law, and that world's access to space and interstellar trade. That Governor is an admiinstrative and diplomatic entity much like the "Customs House" in Colonial America.
One has to wonder for example, whether or not member worlds can go to war against each other. If they cannot, how can they justify the creation and maintenance of a planetary fleet? If wars are fought by men, what keeps the Duke of a subsector from declaring war against another Duke? We have canon material depicting the actions of Dulinor's assassination of the Emperor to the point where he can attempt to take the throne for himself. If this happens at the higher levels of government, why doesn't it happen at the lower levels of government?
You are missing a function of Feudal government. The Imperium is one big feudal hierarchy. Dulinor attempted his attack in MegaTraveller because he was beheading the top and EVERYONE would be his vassel as Emperor(thanks to Olav I, assassination is a legitimate path to become Emperor). So there is no one above Emperor to punish him when he kills Strephon. But every Duke is vassel to the Emperor. So say Duke Lanth gets the grant to administer the Border Worlds. Duke Lunion masses his fleets and squadrons of supporters and attacks Lanth. He may win against Lanth, but now stands in Strephon's judgement.
Remember something very key, Duke Lunion holds the system and duchy in fief from Strephon BUT that is not related to the world government. Duke Lunion can have his title, feifs and Duchy stripped away despite being leader of the Lunion world government. All his possessions as Duke would be removed and awarded to someone else at the gunpoint of the Imperial Navy. But they'd remain King of Lunion and control the world. And a new Duke who would own and live in the Duke's palace and now have control of the Duchy and what military force it owns while Strephon restores(or also replaces) the Lanth Duke.
It is easy to knock off the top if you can control the rest. It is harder to war in the middle levels when there is a boss above you to act. And the further down from the Iridum Throne you go, the more local bosses there are to stop or punish you. A local Baron warring against his neighbor would have to worry about reactions from the Local Duke/Duchess, the Subsector Duke/Duchess, the Subsector Senate AND the Imperial Navy before having to worry about what the Emperor might say in a year or so...
There are so many different possible takes on the Imperium, so much material not filled in, that GM's are in a sense, obligated to determine what is or is not important to their depictation of the Third Imperium and its military, and take it from there.
Much of what you ssem to be missing can be found in earth's own history from the Middle Ages and The Renaissance. There is material enough in the history of the many monarchies that have(and still do) existed. Much like you can use the current Peerage of England as a sort of guide to the structur of local nobility on any world(as modified on a percentage basis by population) in the Imperium, so you can look at the court intrigues and histories to see what being a nobleman is like. Everything from the politically powerless nobles of Louis XIV(France) to the Politically entrenched Nobility of the Russian Courts pryor to Peter the Great. Indeed, in American history before the revolution you have a great many Lords representing fiefdoms in "The Colonies" who did so completely from England as many Marquiees do from the Imperial Moot in the Third Imperium! The parallels are all there and the historys tell you how it all worked(there is a reason that GDW modeled Imperial Nobility on the Victorian Nobility. They ruled much of the world with or without armies, naavies or real local political power assigned to them. The power was always there, as well as the need to justify their actions later, but they did not often have assigned control in and of themselves.
I digress however. The concept of the Duchy navy implies a duchy government, which in turn implies other things. However, HIGH GUARD implied certain things despite it not detailing such things when it discussed the existence of the planetary navy, subsector navy, and imperial navy. Implications however, are not the same as outright assertions of fact (so to speak).
Ultimatly, one must draw from High Guard that any world or other polity that can afford one or more warships to protect their insterests can have said warships...so long as they do not violate the interests and laws of The Third Imperium. That is because once they leave their own atmospheres, they have left their property and are now in Imperial owned space. This results in canon examples like Porozlo, who's attacks on trade brought down the wrath of the Imperial Navy followed by that of the Iridium Throne itself. A Duke/Duchess may have ownership of trade lanes given them as a fiefdom with the understanding they will use the moneis raised from that ownership to create a small fleet of anti-pirate forces to protect trade in that lane(self serving as well as Empire serving) but that Fiefdom(and those forces) can be stripped away at the whim of the current Emperor/Empress as these are..ultimately...the property of the Throne.
I advise you to read a lot more about medieval Europe as you will see many of the bits you are missing implicit in those histories.
Marc