Are there a specific set of educational qualifications for a person to become a Catholic Priest, Bishop, Archbishop, Cardinal, and Pope?
Let us start with the requirements for becoming a Catholic priest. Before going on I would like to state that some requirements may be waived by the appropriate ecclesiastical permission, the local Ordinary or the pope himself.
Another point I would like to mention here, is that priests of the Extraordinary Form of the Mass must have a working knowledge of Latin as their liturgy is entirely in Latin.
Generally speaking it takes eight years of post secondary education to become a priest.
After determining one might like to become a priest one either joins a seminary after high school, or goes to college. A college degree could be in liberal arts or religious studies. Classes emphasizing public speaking and writing are helpful. One can study at either or secular or a religious college. Attending a Catholic college is encouraged since it is thought the college will give the potential priest more access to other priests and opportunities for service in the church.
If upon finishing college, a man still wants to become a priest he will join a seminary. If the man joins a seminary without college, education to become a priest takes eight years. With a college degree, time in the seminary is usually four years.
The timeframe for ordainment can differ from diocese to diocese. Most require that a candidate earn his undergraduate college degree prior to candidacy. The Church requires that priests complete 20 to 24 credit hours in philosophy at the undergraduate level. The seminarian stage requires four years of study in theology at a seminary. After graduation from the seminary, the priest serves for roughly one year as a transitional deacon. It typically takes five years from college graduation to ordainment, provided the priest has studied philosophy at the undergraduate level. If the priest has not done so, an extra year is added to the process, as he would be required to complete a year of pre-theology studies at seminary.
While at seminary, priests take courses in sacred scripture, liturgical theology, moral theology, sacraments, canon law, philosophy and both ancient and modern languages. - How to Become a Priest
The local seminary where I live is basically an eight year program, with an additional years set aside for spirituality.
Our Seminary offers both a four-year program in Philosophy, leading to a B.A. degree and a four-year Theology program with B.Th. and M.Div. degrees for those who meet the graduation requirements. Intellectual formation at SCK draws upon the fonts of Scripture, the Magisterium, Conciliar teaching and St. Thomas Aquinas. Our hope is that our seminarians will be men who think with the Church and are able to appreciate and appropriate into their hearts and minds that which is true, good and beautiful in both classical and contemporary thinking, literature and art.
Human formation at SCK occurs in formal settings such as lectures, conferences and personal direction and in informal ways such as are part of life in community. Students take responsibility for many household chores and share in the work of preparing and celebrating the liturgy. - Rev. Matthew Gerlich OSB, Rector of the Seminary of Christ the King
As for the educational requirements for being a bishop or archbishops, I will let Canon Law speak for itself.
Can. 378 §1. In regard to the suitability of a candidate for the episcopacy, it is required that he is:
1/ outstanding in solid faith, good morals, piety, zeal for souls, wisdom, prudence, and human virtues, and endowed with other qualities which make him suitable to fulfill the office in question;
2/ of good reputation;
3/ at least thirty-Five years old;
4/ ordained to the presbyterate for at least Five years;
5/ in possession of a doctorate or at least a licentiate in sacred scripture, theology, or canon law from an institute of higher studies approved by the Apostolic See, or at least truly expert in the same disciplines.
§2. The definitive judgment concerning the suitability of the one to be promoted pertains to the Apostolic See.
Can. 379 Unless he is prevented by a legitimate impediment, whoever has been promoted to the episcopacy must receive episcopal consecration within three months from the receipt of the apostolic letter and before he takes possession of his office.
Can. 380 Before he takes canonical possession of his office, the one promoted is to make the profession of faith and take the oath of fidelity to the Apostolic See according to the formula approved by the Apostolic See.
Cardinals must meet the requirements set aside for bishops as to be promoted to the cardinalate, one must be a bishop. Seeing that cardinals are named personally by the pope, he alone may waive certain requirements, be it educational or status. Cardinals must be bishops.
Since the time of Pope St. John XXIII a priest who is appointed a cardinal must be consecrated a bishop, unless he obtains a dispensation. These seem to meet the basic requirements set aside for bishops as most are well known theologians and retired rectors of seminaries.
What are the qualifications needed to be elected pope?
Canon 332 of the Church’s Code of Canon Law states: “The Roman Pontiff obtains full and supreme power in the Church by his acceptance of legitimate election together with episcopal consecration. Therefore, a person elected to the supreme pontificate who is marked with episcopal character obtains this power from the moment of acceptance. If the person elected lacks episcopal character, however, he is to be ordained a bishop immediately.”
Canon 1024 states: “A baptized male alone receives sacred ordination validly,” it is concluded that one only needs to be a baptized male in order to be elected pope.
Seeing that Bishops must be in possession of a doctorate or at least a licentiate in sacred scripture, theology, or canon law from an institute of higher studies approved by the Apostolic See, or at least truly expert in the same disciplines, it can be presumed that a pope should have this same intellectual baggage at his disposal, prior to his election. After all, is not the pope, the Bishop of Rome. Needless to say that the Cardinals of the Roman Church could choose a candidate who lacks these requirements, but I seriously doubt they would.