"I haven't got anything" should be accepted, since it is the most common way of expressing this sentence in standard English ('I don't have nothing', however, is nonstandard because of its use of double negatives).
You're right that that should be accepted but I think that "I don't have anything" would still be more common.
The fast speaking version sounds like nu and am have been contracted. Therefore shouldn't 'eu n-am nimic' also be accepted? Or is this not correct Romanian?
Eu, eu, care nu am nimic!
How do you write: I have not nothing! ?
You don't. English doesn't use double negatives and it's considered grammatically incorrect. So the correct sentence is "I have nothing".
"I have not nothing" doesn't make sense but it would mean "I have something"