I suspect that the non-subsector dukes are more like combined Taxcollectors and Adjutant Inspectors-General than rulers. Still, I've always allowed them to be rulers of the starports.
I suspect, based upon the JTAS 14 article
high justice that subsector courts are filled with justices who are nobles, if not honor peers, overseen by the Subsector Duke and formally appointed by the Sector Duke on the recommendation of the Subsector Duke. I have always had the same general set of crimes as JTAS 14 punished in Subsector Courts, with Nobles being able to be heard by the Subsector Duke.
Crimes normally subject to subsector justice include the unlawful introduction of high technology to low technology worlds, the violation of a planetary interdiction, possession and/or use of nuclear, chemical, or biological weapons, and war crimes as defined in the Imperial rules of war.
(JTAS 14, High Justice, p.20)
Also interesting is the nature of "high courts"...
The prosecutors and high courts of the lmperium concern themselves with crimes that would affect worlds in more than one subsector of the realm. These include the capture, transportation, and possession of slaves, piracy, the murder of Imperial officers, officials, or members of lmperial nobility, theft of lmperial property, treason, and conspiracy to commit treason against the lmperium (note that treason against a planetary government is a local crime).
(ibid., p.20-21)
I've always put those tried at the subsector level, and appealed to the Sector and Domain levels. Guess I've been wrong about those for 20+ years.
It also notes about bribery that bribery is harder than baseline in a subsector one and "impossible with an Imperial Tribunal". The required skill for a subsector attorney is Admin 3, and for "high courts" it is admin 4. (this was an '82 publication - legal skill is in the 1985 Merchant Prince).
Not stated, but congruent with other canon, would be a requirement that Subsector and Sector courts require a knighthood or peerage to serve upon.
Likewise, I suspect the subsector dukes and local nobles should have inspection authority for local imperial offices. Rather than direct oversight, or hire/fire, they should be able to inspect and investigate.