Dozens of bishops serve the Holy See in various capacities as Curia officials or as Nuncios / special representatives to nations and international bodies around the world. While all are bishops, none are ordinaries of their own particular churches (e.g. Dioceses). Vatican II (and the catechism) clarified that the bishop is truly head of his own particular church and a vicar of Christ, not a mere vicar of the pope, even if he is subject to the Pope in as far as he exercises his universal jurisdiction for the good of the Church. These dozens of Curia / nunciature bishops are, on the other hand, the pope’s vicars, exercising some aspect of his universal primacy. As the pope’s office is derived from the fact that he is Bishop of Rome, could it be argued, theologically, that all of these bishops are essentially the equivalent of the Roman Church’s auxiliary bishops?
No
The Roman Church, assuming you mean the Diocese of Rome, has eight auxiliary bishops and the Cardinal Vicar who governs the diocese day to day
The Vicar General for Vatican City is also a Cardinal…but I remember the days of the very long serving Bishop who was Vatican City’s vicar general. Having been very long serving, he is also very long gone. Vatican City is on its fourth Cardinal Vicar in succession after him. There are no auxiliary bishops for Vatican City but their function is essentially supplied by those Canons of Saint Peter’s Basilica who have received the episcopate
The analogy of the relationship of a diocesan bishop to his auxiliary bishops really does not apply to any of these situations…except for the actual auxiliary bishops of the Diocese of Rome. In a real sense, the Cardinal Vicar for the Diocese of Rome and the Cardinal Vicar for Vatican City would have a relationship to the Pope that is analogous to that of an auxiliary bishop to the diocesan bishop…but practically speaking, it really isn’t so on a day to day basis. Here it is very nuanced. It is a unique situation, ecclesiologically
In the dicasteries, the prefects of the Congregations are each Cardinals. Until the present moment, one who is named to head a Congregation but who has not yet been elevated to the dignity of Cardinal would bear the title of “pro-prefect” pending his elevation to Cardinal. The Cardinal Prefect would typically be received in audience by the Holy Father once per week for conducting necessary business…this would include the Prefect for the Congregation of Bishops, the Prefect of the CDF, the Prefect for the CDWDS and so forth
The number two official, the Secretary, holds the rank of Archbishop
There are offices which have lesser oversight. The Prefect of the Prefecture for the Pontifical Household, for example, is currently an Archbishop
You are correct that dicasterial functions are purely derivative – they depend upon the authority of the Pope and if the See of Rome falls vacant, their functions are suspended…except for the Dean of the Sacred College, the Camerlengo of the Holy Roman Church and, for reasons of practical necessity because it directly concerns the salvation of souls, the Cardinal Grand Penitentiary of the Holy Roman Church, for emergency resolution of cases involving the internal forum
So there is some similarity but there are also very distinct differences. And the actual interaction is quite distinct within the Vatican curia from what would happen in a diocesan curia…structurally but also canonically and theologically
As for the diplomatic corps, that serves a dual function. As a sovereign Head of State, the Pope exchanges diplomats with countries that form the family of nations with which diplomatic relations have been established
The nunciatures, which are our version of embassies, have functions relative to the civil government they have been sent to. Nuncios would present their letters of credence to the Head of State just as every other ambassador does. There is a similar structure with a deputy chief of mission (chargé d’affaires) and so forth
Within the corps, similarly, there is a ranking in terms of assignment. Those who are nuncios will be archbishops. Officials of the next level will be
monsignori with the title of protonotary apostolic. (There are some posts where there is a chief of mission who does not have the rank of nuncio and they will be protonotaries apostolic.) The next will be
monsignori with the rank of Prelate of Honour. The next will be
monsignori accorded the rank of Chaplain of His Holiness. Finally priests in their first assignments out of the Pontifical Ecclesiastical Academy may not have yet been elevated to monsignor
In these capacities, the nuncios are representing the Pope as a Head of State and functioning as an ambassador with guidance coming from the Secretariat of State
The nuncios also have the function of representing the Pope to the bishops of a given nation. The nuncio will be present and engaged, for example, when the country’s national conference of bishops will be meeting. He will be present, normally, for the ordination and installation of bishops, as the envoy of the Pope, and will typically have a role also in those ceremonies. He is also involved in the proposals going to the Holy See for nomination of bishops, which is ultimately the personal prerogative of the Pope
Thus his rank as archbishop is also very important in this role since he is the equal of the nation’s metropolitans…however granting that possibly one or more metropolitans (and even non-metropolitans) may have been granted the dignity of Cardinal. It is also in this capacity that the nuncio will be interacting with, for example, the Cardinal Prefect of the Congregation of Bishops
You are correct to ascribe a special relationship of the diplomatic corps members to Rome…but more than the Diocese of Rome, it is to the Papal Court and to Vatican City since they hold Vatican passports