How do cases in German work when with other words?
I know that you use cases when saying things like "Ein Auto, Das Auto..." but if I wanted to say that the car is mine, do I say "Mein Auto" or "Mein des Auto" or just "Des Auto", and why?
There are other better places to learn about cases, so I'll just answer your specific question.
'Das Auto ist mein.' = 'The car is mine.'
'Das Auto' is in the nominative case because it is the subject. 'mein' is a possessive pronoun. It is also in the nominative because the verb is making an equivalence between 'Das Auto' and 'mein', so they have be the same case.
'Mein Auto' means 'My car'
'Mein des Auto' is incorrect whichever way you look at it. 'Des Auto' would have to be 'Des Autos' and means 'of the car' so it would have to follow another noun. Ex. 'der Fahrer des Autos', 'the driver of the car'.
the most literal translation is
Dieses (This one) meiner (out of my) Autos (cars) ist Schrott (is junk), which not only includes a cumbersome Genitiv construction, but also implies that you own several cars. The english phrase This car of mine technically also implies this, but it would also be acceptable if you only own one, because it has kind of developed into an idiomatic expression.
So, there is no good way to say that in German, you would just say
Mein Auto ist Schrott or Mein Auto hier ist Schrott