Few things I've discovered

>Everstone still gives a 100% chance to pass down nature.
>If mother has one of two normal abilities, child is still much more likely to inherit that ability.
>Child hatches in the same type of Pokéball as the mother.

I've seen reports fathers can pass down Hidden abilities now, but after catching a Swift Swim male Poliwag, all I've been able to breed is a box of Water Absorb Poliwags, so maybe there's another condition I'm missing.

Since I have nothing better to do right now, if anyone wants to suggest a move and a Pokémon, I can try to test if that mon gets that move via breeding.