This is a cultural city does not make sense in English. What you would say is that a city is cultured, ie a city where culture flourishes.
Agreed. "This is a cultural city " is the sort of nonsense Google Translate provides for us to scorn .
Is there a trick to help remembering when such words begin with c (as in English) and when with k (as in German)?
It seems odd to me that Dit can be both singular and plural, this and these. Have i misunderstood?
Yes and no :) As a dependant demonstrative pronoun dit (this) is singular (e.g. dit meisje - this girl, meisje is a het-word), and is used for het-words. Singular de-words use deze for this (e.g. deze stad - this city, stad is a de-word). Plural nouns are always de-words and use deze (these) (so "de meisjes, deze meisjes, de stad, deze steden - the girls, these girls, the city, these cities).
However, as an independant demonstrative pronoun dit is used as both this and these and can be singular or plural as you suspected (e.g. "dit is een grote stad" - this is a big city, "dit zijn grote steden" - these are big cities).