Since "Não existe nada [...]" was not accepted, are double negatives not used?
Yes! Using "nada" sounds much more natural :)
Can someone please explain why the English adjectives "independent national" are ordered as "nacional independente" in Portuguese?
I think this is just a specific choice of word order and not a general rule. As in a national newspaper which is independent as opposed to an independent newspaper which is national.
It accepted both word orders for me.
Why not - Não existe uma coisa asim como um jornal nacional independente?
That is not a sentence you'd hear from a native.
Is "não existe nenhuma coisa..." also incorrect? That is what I tried.
I can't remember the last time I heard "nenhuma coisa". It's not common. People use "nada" instead.