I'd like to know if there's any relevant distinction among German newspapers with respect to the sophistication of the use of language (both syntax- and vocabulary-wise). I've seen in numerous websites that FAZ, ZEIT and SZ are the most renownd German newspapers and I've seen numerous comments on their respective political tendencies, but I've yet to see someone comment on the quality of the use of the German language in each of these.
The originality of the die Tageszeitung (die taz) would make it a little bit more demanding than the average.
If you are looking for extremes - and "most demanding" suggests you are - than try reviews of classical concerts in any of the big papers, and (IMHO) in particular in the FAZ. But don't take them as examples. It is a special style that would sound outright ridiculuous in any other context. Sometimes it feels just like a competition in inventing words and phrases.
The ZEIT and the Süddeutsche Magazin both have crosswords that are largely based on play of words. Probably quite demanding for a non-native speaker, too.
As for the "normal" texts, in my opinion the differences among good and bad texts (from a language perspective) in any one of the papers you mentioned (plus WELT and SPIEGEL) are bigger than the differences of the average level (whatever this should be) of these papers. So you can safely select them by their quality of journalism (which is out of scope here) and will get more or less the same language level.
If you want to read well-written pieces (both in terms of style and journalistic genre), I recommend having a look at Page 3 (Die Seite Drei) of the Süddeutsche Zeitung daily.
There you find (not every day but very often) true masterpieces of journalism. Not standard stuff from the journalistic word mill, rather things that focus on background and are written to last as literature; things that can be amusing and enlightening to read even if you are not interested in the actual topic of the narrative.
Some of the authors have also published books with their "Seite 3" pieces. My favourite was always the old master of ironic reporting, Herbert Riehl-Heyse, who unfortunately died, much too young, in 2003. I recommend especially:
Herbert Riehl-Heyse: Die Weihe des Ersatzkaisers und andere Geschichten. Basel 1986.
You find this book currently at prieces starting from 1 euro at many second hand book traders. And here is a link to one of the pieces from the book: http://reporter-forum.de/fileadmin/pdf/Gern_Gelesen/riehl-heise_wortgekringel.pdf (link checked 2019-02-26).
Herbert Riehl-Heyse: Das tägliche Gegengift. Reportagen und Essays 1972 - 2003. München 2008.
Herbert Riehl-Heyse: Bestellte Wahrheiten. Anmerkungen zur Freiheit eines Journalistenmenschen. München 1989 (Kindler).
PS: I read the article about the Klagenfurt journalist award right now again. Yes, this is indeed demanding. You need, I believe, a very high level of understanding of German, including text genres, and various registers of speech, in order to feel the irony in it. It sometimes hides in nothing else but the way a sentence is built.