Questions & Answers about Jakiego koloru jest twój samochód?
Why is it Jakiego koloru (genitive) instead of Jaki kolor (nominative)?
In Polish, when asking about a specific trait like color, you often use the genitive case to mean "of what color". So, instead of asking "what color is the car", you are literally asking "of what color is your car?". This is why both jaki and kolor take their genitive forms: jakiego koloru.
Could I leave out the word koloru and just ask Jaki jest twój samochód?
You could, but the meaning changes. Jaki jest twój samochód? translates to "What is your car like?". The person might answer that it is fast, old, or broken. If you want to know the color specifically, you must include koloru.
Why don't twój and samochód also change to the genitive case here?
Because twój samochód (your car) is the subject of the sentence. It is the thing that "is" a certain color. The subject of a sentence always takes the basic nominative case, regardless of what case the describing words at the beginning use.
When I answer this question, do I need to use the genitive case for the color?
No, you don't have to. While you could mirror the question's grammar and say Jest czerwonego koloru (It is of red color), it is much more natural and common in everyday Polish to just drop the word for color and use the basic nominative adjective: Jest czerwony (It is red).