'Old-fashioned' is interesting.
In this context, old-fashioned is definitely a describer because the 'and' wouldn't make sense if old-fashioned were meant as a classifier. Compare these:
A quiet and old-fashioned place... (makes sense)
A quiet and police place... (doesn't quite make sense)
A quiet and French place... (doesn't quite make sense)
But if the intention was to classify things by their date or style, then old-fashioned could become a classifier, you just wouldn't use 'and'. For example:
A quiet old-fashioned place...
A quiet modern place...
The key difference here is that describers represent subjective opinions, whereas classifiers represent objective categories.
'Old-fashioned' could be either: it could be a subjective opinion or it could be a categorical classification.
But in this snippet, it works as an opinion.