Whereas i seriously pondered "the key of the door", since it uses a definite article Yes it would mean a specific door, but i cant see how one can make that distinction from the polish A key to a door, any door, where the type of key is in question The key of the door, when i specifically want to lock/unlock a given door Does that help?
The first word would be "kluczyk", it's a diminutive form, indicating that it's a small key.
Beyond that, there are various ways to call it. "kluczyk do samochodu" (samochód needs to be in genitive, and drzwi are too in this example, but it's not as easy to notice) is one way. You could also say "kluczyk samochodowy" (samochodowy is an adjective). But in my home we call it "kluczyk od samochodu", like Russian, but I don't know if it is accepted in the standard language.