Answer by Gérard Lang for Order in bijective-equivalent collections of proper classes in set-theory

The answer to question 2 is Negative. This is because, answering to my question "Injection of the proper class of ordinals into every proper class", J.D. Hamkins proved on 02/12/2014 the existence in NBG of a proper class W that does not inject into On.He also proved that On does not inject into W.So that if A were to be the minimal proper class for injection into proper classes, A would inject into On, so would then be bijective with On and well-orderable. But A would also be injective into W; but by chaining an injection from On into A and an injection from A into W, we would get an injection from On into W, that is impossible.Gérard Lang