If someone has "a red nose", it may means he drinks much alcohol.
I've heard of this happening once in India.
Or he was too long in the African sun.
it could be worse if he or she was a regular drunkard—then, the nose may be red, swollen and veined. Maybe there is a word for it in Polish too; in English, the only translation offered to me simply was “a drunkard's nose”.
Is 'czerwony nos' accusative? Do you always use accusative when using 'I have a /body part related thing/?'
You simply use Accusative after "mieć" (to have) :) Yes, it is.