The Welsh here is correct - as you will be realising by this stage of the course, Welsh and English do not use the same language patterns. The double negative is the usual pattern in this context in colloquial Welsh.
Isn't it saying no, i have no car? I don't see three negatives that Ian's comment implies?
The confusion is probably that this particular pattern splits the 'does' and the 'dim'
Does dim are usually paired to express 'there is not'
In South Wales the pattern is clearer = 'Does dim car gyda fi' = (lit:- 'There is no car with me' ) 'I have no car'/I do not have a car.
A more formal pattern in the North is 'Does dim car gen i', which again makes it clearer.
The pattern above is the common pattern in spoken Welsh in the North.