Oof, yeah, that's a pickle. I mean, if I try to explain that without much thought, I fall back immediately on Russian, where those categories are... well, even more complex.

Although you do have an error in the second sentence. It's "Einen Hund hat meine Freundin."

Quite frankly, I can't come up with a context in which it would make sense. But, I think it would go better with the definite article. "Den Hund hat meine Freundin." So, "der Hund" has already been established. In this sentence you're conveying who exactly has the dog - "meine Freundin". Which is in the nominative case, whereas the dog is in the accusative (a direct object), although I imagine that doesn't help a lot. Or, following on from our earlier discussion about the multipurpose nature of the definite article, you could kind of reverse the sentence in the translation - "That (particular) dog belongs to my girlfriend." ...Let me now if that's even more confusing.