I think he is talking about Soma's father and just the Nakiri family in general. I don't won't to say too much, as I think it will be revealed soon in some of the upcoming episodes. 

I will be honest, I think Erina and Soma's mother are the same person and perhaps she died. At least that is what I believe. 

Probably against the "competitive" culinary world for whatever it ruled on Joichiro. Azami wants to eliminate shokugeki's and expulsions altogether because he believes that to be an injust world where potential is lost but this ideal is irreconcilable once we consider that potential can only be compared to one or another. Central was created in the first place to define what the perfect dishes are but as we see upset victories against the body, it is failing in that respect ... 

Azami wants to create this so called "safe space" or "echo chamber" where Joichiro's original cooking can flourish ...

A world without any confrontation is weak and Totsuki doesn't need the current Elite Ten who won't try to even reach new heights and especially Eizan who lost to a total dunce like Takumi in which he was only focused on growing his business instead of his cooking ... (Shokugeki's should be the ONLY criteria in the future for consideration of an Elite Ten seat if these students are to represent Totsuki's best chefs, everything else was a mistake) 

The Elite Ten is weak because they aren't honing their current culinary skills and will most likely lose to the rebels ... (hopefully after seeing their own error and if they still have any sense left they'll see to it that dissolving Central was the right thing at the end)