Imho translating names should be also correct, optionally. It is quite common to officially change your name to the version of the foreign language after all. German Franz becoming English Frank, Schmidt becoming Smith. We would learn which are related and which have no English counterpart.
For individuals I would not do it though. So just as an excersize here.
Queen Elisabeth, the Austrian-Hungarian one, is Erzsébet for them. I would not be surprised if Hungarians do not know "Elisabeth" when they don't speak any German. At most associate it with the British queen.
Names of places are like calling Hungarians Hungarians although they call themselves magyars ... Each language decides if they adopt the foreigners own nomenclature or if they already have made up something on their own for sometimes odd reasons.
I would believe there are some historical reason why this happened. Budapest being Budapest everywhere (?) but English Vienna being at least similar to German Wien while Bécs is something else and English/Slovak Bratislava (formerly it was Prešporok) / German Pressburg (although it is basically Bratislava now for us too) being Pozsony.
I have problems to distinguish k and g in Hungarian. The k is supposed to be "softer" compared to the German k, but that makes it immediately sound like a g to me.
Göszönöm, Gati, Gatika, Garcsi. Or garázs. ;-) I always would have thought it is written like that until I started to learn Hungarian.
Weird. In speech I probably will just make my "normal" k sound but I have no idea how I should hear k differently when it just sounds like another letter to me.
In German, syllable-initial voiceless plosives (i.e. 'p', 't', and 'k' at the beginning of a syllable) are aspirated, which means that they are formed with extra pressure, producing a little puff of air when spoken. But we also have non-aspirated ones. Compare the pronounciation of "Teig" (with aspiration) and "Steig" (usually without aspiration, where the 't' sounds closer to a 'd'). Or "Pannen" and "spannen". Hungarian consonants are generally unaspirated, which makes them sound softer.