I was first curious to how Kurama just happened to have decent parenting skills, but then I heard that in myths kitsune were known to be doting wives and good mothers!
Kurama's behaviour - reckless, hates to be pitied, prone to cause trouble - is often compared to Kushina's and the Uzumaki Clan's in general. Of course Kurama acts as a Uzumaki - his three hosts were Uzumaki, meaning that he learned human behaviour from them and subsconciously mimics them.
When Naruto died, his body transformed to reflect his new occupant. The Kyuubi was a red fox with red eyes, and Kurama is a redhead with red eyes.
Kakashi's complete avoidance of Naho in the first few chapters seems funny, but then you realize that this is why he never approached Naruto even after he was older and (somewhat) more mentally stable. If he was this inept in such a short period of time, a young child with serious emotional and physical needs would be completely beyond his capabilities - and he knows it.
