"My son's wife does not have a car, but she has a bicycle."
Translation:La esposa de mi hijo no tiene un automóvil, pero tiene una bicicleta.
would 'la esposa de mi hijo no tiene un coche, sino una bicicleta' also make sense?
They should accept nuera as an answer. Mi nuera no tiene coche, pero tiene una bicicleta.
duolingo keeps suggesting correct translations of "a car" as "coche" in stead of "un coche"; what´s up with that? literally it says: "... does not have car". Is it common slang, or official grammar? Is it only applicable to car, or only in negative sentences?
If you say "un coche" might mean that she has not only a single one, but instead it might be that she has none or several cars, so it is better to use "no tiene coche" instead "no tiene un coche".
If you want to use an article, use, ningún instead of un. Otherwise just skip it altogether, like Hikaru says.
if you say "mi hijo's esposa" would this make sense to a native speaker? Or would you just sound crazy?
Lol I wouldn't recommend it. It would sound very strange and unless they knew a good amount of English, the wouldn't understand it. You could try though! Check out the looks you get :)
As Iago says... no chance. I wouldn't recommend it at all. Possessive in Spanish is rather easy though: article + object possessed + de + owner.