"our dog" is just the subject of the sentence. So as a whole it takes Nominative.
Don't go too far with associating possession with Genitive. It's not that every single word connected with this topic takes Genitive. What we mean is that it works in constructions like "Josh's dog" -> "pies Josha". Which after all is called 'Saxon Genitive' even in English :)
mały - masculine, right.
mała - feminine, right.
mali - it's 'masculine personal' plural. You can remember that this form is usually quite different from the other ones, kinda softened.
The neuter one, as well as the 'not masculine-personal' plural one are both "małe".