This does kind of start to adopt a counter-accusative tone, which could be interpreted as negative. Though it would entirely depend on context. The expletive just makes it sound angry overall.

Technically this isn't a double negative, but you are right that the negative connotation of "hate" is reversed by "don't".

"I don't have nothing" is a double negative. In "proper" English (whatever that means ;p) , the correct phrasing would be "I don't have anything" or "I have nothing".