He has a special lineage that probably made him have Conqueror's, like explained before in Luffy's case.
As for the swords, I already explained it. Enma taught him how to flow his haki and unlock KoH. Wado helped him use Enma's full power since they were forged by the same person. Without them, would Zoro have defeated King ? I do not think so.
Conqueror doesn't work like that. Is about the individual and not about his family.
Katakuri so far is the only on Big Mom's family that has Conqueror.
Don Chinjao son doesn't showed Conqueror yet.
Nobody besides Doffy in his family showed Conqueror.
Neither Momonosuke or Hiyori showed Conqueror yet.
Hancock's sisters didn't showed Conqueror yet.
Enma didn't taught Zoro nothing. That is headcannon. Zoro learned Ryou at Alabasta fighting Mr 1 because his village master once told him about breath of all things. CoC coating was unlocked in an extreme situation just like Luffy did after being knocked out by Kaido again. Rayleigh let that clear when he is trainning Luffy. Haki blooms on extreme situations. There is no magic way to unlock haki, advanced versions etc.
Zoro had better chances against Yonkos and King without Enma. He didn't even needed CoC coating if he had Shusui in the place of Enma.