If faction A changes their name to "Faction B," do they still have any rights or say about how the name "Faction A" is used, since anyone could start a new faction and name it "faction A" since the name is unused?
If a faction drops their name and takes a new one then they do not have any rights to their previous name. Any faction can take the dropped name and do as they want with it. As for your description you should be able to use any dropped faction names that you wish, even if the faction that used to have that name disagrees.
Can I make any claims about Krath Dynasty, now that the faction dropped Dynasty from its name?
Nothing personal against Krath in whatever version, it's just one of the most recent prominent renamed govs.
Also most factions that have been taking on names of old factions have done some work to differentiate themselves from previous iterations in one way or another while still laying claims on the name. Otherwise any grievances or contracts made with the previous iterations would transfer to the new ones, would they not?