mit verletzten Mund und Nase
This doesn't work. Here you are using verletzt in plural dative. Since in German adjectives have to agree with their noun on number, case and gender, there must be something in plural that verletzten can refer to. Mund und Nase, however, do not form a plural entity (at least not to my ears). The same holds true for Hemd und Handtuch, so you second example doesn't work either.
There are rare cases where this does work, e.g.
die berühmten Siegfried und Roy
In this example the names of the two artists appear like the name of a group that can be used like a plural word.
mit verletztem Mund und Nase
as it was suggested in another answer, might work – but only because "mit Nase" alone doesn't seem to make sense at all, so that a reader might accept the error in agreement (Nase is feminine, while verletztem is masculine) in exchange for more reason. But the error still lingers around, making this phrase sound wrong somehow. Compare
mit gebrochener Nase und Sonnenbrille
Nobody would think of broken sun glasses here, even though agreement rules would be obeyed. And that's not only because one would use "zerbrochen", not "gebrochen" for glasses.
Ich esse gerne frischen Fisch und Käse.
Again, frisch is only applied to Fisch – Käse stands on its own and is not described in detail.
kluge Jutta und Ela
It's similar to the Fisch/Käse example, so I'd say it's option 1) – and 4), because I cannot figure how someone would use it without an article. But maybe someone might come up with a context where this phrase, without article, does sound natural and can be understood in the way of 2), so you could also say it is 3) ;-)