Everything is pretty much right as far as I remember, a few small tweaks, Rina was actually fiancee of Dante at the time of that photo, and I think (it's not confirmed) Don Silverino was fiancee of Cordia. Rina and Don Silverino did betray Cordia, and MC is indeed their son.
As for who Antonio's parents are, I am 99.99% sure it's Lumina (Younger sister of Cordia, Cordia killed her) and Don Profaci (he is one of the big5), why do I think that? Well codex makes it clear he is NOT Cordia's son, and in the pile of old contracts there is the name 'Antonio Profaci', and because it also states that eldest son of Don Profaci was taken hostage at the end of War.