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.

"I don't fucking hate dogs!!" sounds like something yelled with a raised fist-finger in a dispute about dog shit on the lawn between neighbours in some lowly apartment complex.

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".