It's complicated, but generally the answer is 'jein'.
Majestätsbeleidigung is on the books, along with Gotteslästerung and Staatsfeindliche Handlungen. It was used widely, some would say indisciminately, in the 20s and 30s, a sour memory for German jurists. But by the 2000s, the law is circumscribed heavily. Firstly, the penalties were never all that stiff. Up to five years of Festungshaft was theoretical,in reality you would be spending a few months in comfortable impriosonment and the publicity was worthwhile (Festungshaft was 'nicht ehrenrührig', you did not come out a felon, and the Reich was civilised about it). Secondly, the law requires the victim (the sovereign or relative or their guardinan) to authorise prsecution. It is not a good idea to do that too often, not least because you can actually lose those cases. And thirdly, because under German law, truth is an absolute defense in libel cases and over time, more than one sovereign got burned prosecuting the libel of a minor relative that way.
Of course there are real limits to what the press gets to say even in modern-day Germany. there are libel laws, privacy laws (though these are much weakth than IOTL without the experience of Gestapo and Stasi), and a myriad of ways that the powers that be can make your life uncomfortable if you put the wrong stuff into print. But it's not really much different from the way the press works in modern-day France or Germany. and the strong, but highly circumscribed libel laws also have the side effect of curbing the worst excesses of conspiracy theory. There is no equivalent of Whitewatergate or Birtherism ITTL. Any recognised, licensed media outlet peddling that would be sued out of existence in short order.
Technically yes. In fact the armies are entirely integrated into the unified military structure. All regiments are still formally affiliated with one or more member states, but the only ones still under a command outside the Great General Staff are the guards units. These remain under the direct command of the sovereign and range in numbers from corps strength (Prussian and Bavarian lifeguards) to thirty (Hamburg's Reitendiener). In wartime, they are put under unified command via a process analogous to the federalisation of National Guard units. And of course the navy and air force are completely centralised under the emperor.
All member states maintain embassies and consulates abroad to further their own interests, but the whole thing is basically a patchwork of overlapping cooperative agreements that it might as well be called 'German'. Basically, any consulate of a member state can do administrative work pertaining to any other. They have agreements to assist each other's citizens. that is why, for example, Prussia has no diplomatic presence anywhere in Oceania. Hamburg has that covered. Generally speaking, Prussia's diplomatic service is the most important one (and the one affiliated with the foreign intelligence service), so in most cases the Prussian embassy also 'handles' the German ambassador. Only Bavaria and Saxony bother to have any embassies for reasons other than traditional (closing an embassy would be a needless snub to a foreign power, and ambassadorships make nice sinecures), but many states have consulates. These mostly serve economic purposes. Bavaria, for example, has a consulate in Warsaw whose job is to make sure that Bavarian companies don't get left out of lucrative deals. That is how these things get done.
/pic161468.jpg)