Are there any other examples in medieval Europe (Iberian peninsula, Britain, Scandinavia, Eastern Europe etc.) when initially administrative positions (appointed by king) became hereditary like in Frankish kingdoms counts gained hereditary rights over their counties during the 9th century?
The only other example i know is during the late 13th century in kingdom of Hungary - the counts (hun - ispan, lat - comes) receiving grants or outright usurping the hereditary rights over the counties. At the same time old native Arpad dynasty died out. And the new king of Anjou dynasty had to fight long "civil" wars to overturn those privileges and grants.
Well, sticking within the Frankish kingdoms, in 11th C. northern Italy, there was an attempt by lower lords (
valvassores, hitherto appointed by counts) to acquire the same hereditary rights as their superiors. It was initially resisted, but Conrad's
Constitutio in feudis of 1037 granted their claims (within the Kingdom of Lombardy, it did not extend to the Papal States or elsewhere). So I guess that is a secondary (but derivative) example of the above.
Otherwise, there weren't that many administrative positions in Medieval kingdoms to begin with.
Some administrative positions not tied to landholding were positions in royal households and armies which started as appointments became hereditary - but at the same time they also lost their power and became merely symbolic or honorific. The obvious ones are the myriad of cup-bearers/sword-bearers, etc. e.g. the HRE electors all had hereditary administrative positions in the imperial household, but they meant nothing but symbolism & rank - cup-bearer (Bohemia), chamberlain (Brandenburg), marshal (Saxony), steward (Palatinate, later Bavaria), treasurer (Palatinate, later Hanover), etc. Similarly true in many other countries as well. The "Lord Great Chamberlain" of England started off as real, but became hereditary in the Earls of Oxford, the "Earl Marshal" became hereditary in the Dukes of Norfolk, the "Admiral of Portugal" became hereditary in the Pessanha family, etc. I'm sure you could come up with examples like that for everybody. But as soon as they became hereditary, these positions lost all their real power.
At a stretch, you can say "Bishoprics" became hereditary during the Protestant secularizations, but of course, the position was no longer the same position - an "administrator" of an ecclesiastical state did not have the spiritual jurisdiction the bishop had before.
Taxation did not begin until 14th C., and given their "extraordinary" (intermittent) nature, their collection was never regular enough to make it hereditary. These are distinct from "ordinary" revenues from royal property, tolls, dues, etc. Many of the latter were initially administrated by royal officials (bailiffs, senechsals, etc.), and many of these too became hereditary - but not in themselves, but passed permanently to local counts or towns at some point, and thus attached to their original grants. Some that didn't pass were farmed out, and I wouldn't be surprised if in some places that tax-farming contracts were de facto hereditary, renewed with the same family.
Other judicial positions, e.g. judges in royal courts were never hereditary - for obvious reason. The whole point of a royal court was to sidestep/supersede the hereditary jurisdiction of the count's court with appointments controlled by the crown (& allowing the position to be sold & resold to raise new revenues for the crown). But I am sure in some places, royal magistrates slipped into hereditary for a few generations.
