Is an individual bishop in communion with the Church and Supreme Pontiff prevented from teaching/approving heresy via formal acts under the doctrines of Infallibility or Indefectibility?
According to Catholicism, do individual bishops have the safeguard of Infallibility or Indefectibility?
The short answer on both counts is no.
First of all what is indefectibility?
Imperishable duration of the Church and her immutability until the end of time. The First Vatican Council declared that the Church possesses "an unconquered stability" and that, "built on a rock, she will continue to stand until the end of time" (Denzinger 3013, 3056). The Church's indefectibility, therefore, means that she now is and will always remain the institution of salvation, founded by Christ. This affirms that the Church is essentially unchangeable in her teaching, her constitution, and her liturgy. It does not exclude modifications that do not affect her substance, nor does it exclude the decay of individual local churches or even whole dioceses.
By definition the local ordinary of a diocese does not enjoy the safeguard of indefectibility since it’s definition states clearly that it is accorded to the Church in its entirety. It is not the pope or a bishop that is indefectible, but the Holy Catholic Church itself. Whole dioceses can vanish and no longer exist. Historically, this has already happened many times over. However the Church will last until the consummation of the world.
Dioceses that no longer exist are generally referred to as titular sees.
A titular see in various churches is an episcopal see of a former diocese that no longer functions, sometimes called a "dead diocese".
The ordinary or hierarch of such a see may be styled a "titular metropolitan" (highest rank), "titular archbishop" (intermediary rank) or "titular bishop" (lowest rank), which normally goes by the status conferred on the titular see.
The term is used to signify a diocese that no longer functionally exists, often because the diocese once flourished but the territory was conquered by Muslims or no longer functions because of a schism. The Greek–Turkish population exchange of 1923 also contributed to titular bishoprics. The see of Maximianoupolis along with the town that shared its name was destroyed by the Bulgarians under Emperor Kaloyan in 1207; the town and the see were under the control of the Latin Empire, which took Constantinople during the Fourth Crusade in 1204. Parthenia, in north Africa, was abandoned and swallowed by desert sand.
Do individual bishops have the safeguard of infallibility?
This part of the question needs some clarification before going on. On their own, an individual bishop does not enjoy the privilege of infallibility. Only the pope and only when he speaks ex cathedra—that is, when in discharge of the office of pastor and teacher of all Christians, by virtue of his supreme apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the divine assistance promised to him in Blessed Peter, is possessed of that infallibility with which the divine Redeemer willed that his Church should be endowed in defining doctrine regarding faith or morals; and therefore such definitions are irreformable of themselves, and not in virtue of consent of the Church". Most bishops never teach erroneous doctrine and many are saints. But nevertheless they do not enjoy the privilege of infallibility in their episcopal office. It belongs to the Successor of St. Peter the Apostle.
Freedom from error in teaching the universal Church in matters of faith or morals. As defined by the First Vatican Council, "The Roman Pontiff, when he speaks ex cathedra—that is, when in discharge of the office of pastor and teacher of all Christians, by virtue of his supreme apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the divine assistance promised to him in Blessed Peter, is possessed of that infallibility with which the divine Redeemer willed that his Church should be endowed in defining doctrine regarding faith or morals; and therefore such definitions are irreformable of themselves, and not in virtue of consent of the Church" (Denzinger 3074).
The bearer of the infallibility is every lawful Pope as successor of Peter, the Prince of the Apostles. But the Pope alone is infallible, not others to whom he delegates a part of his teaching authority, for example, the Roman congregations.
The object of his infallibility is his teaching of faith and morals. This means especially revealed doctrine like the Incarnation. But it also includes any nonrevealed teaching that is in any way connected with revelation.
The condition of the infallibility is that the Pope speaks ex cathedra. For this is required that: 1. he have the intention of declaring something unchangeably true; and 2. he speak as shepherd and teacher of all the faithful with the full weight of his apostolic authority, and not merely as a private theologian or even merely for the people of Rome or some particular segment of the Church of God.
The source of the infallibility is the supernatural assistance of the Holy Spirit, who protects the supreme teacher of the Church from error and therefore from misleading the people of God.
As a result, the ex cathedra pronouncements of the Pope are unchangeable "of themselves," that is, not because others in the Church either first instructed the Pope or agree to what he says. (Etym. Latin in-, not + fallibilis; from fallere, to deceive: infallibilis, not able to deceive, or err.)
If individual bishops in union (at the same time) with the Supreme Pontiff were to make an infallible teaching within the Catholic Church, they would enjoy this privilege, uniquely by virtue of being united to and in union with the pope at that exact moment would they be infallible. Apart from this unique perspective, generally speaking, bishops are not infallible. They can make theological errors.
A bishop in full communion does not have the grace of infallibility. Bishops do not have the grace of infallibility, that is they do not have the safeguards of the assurance that they will always be infallible.
Thanks be to God, the vast majority do not.
It should be noted that bishop who writes a theological work must still obtain an imprimatur. He can not grant an imprimatur to his own works! Papal encyclicals do not carry an imprimatur! There is no need to do so.
Bishops "infallible when in agreement with the Roman Pontiff they impose on the faithful a doctrine to be held definitively".
Sacræ Theologiæ Summa IB p. 198:
Bishops, successors of the Apostles, are infallible when in agreement with the Roman Pontiff they impose on the faithful a doctrine to be held definitively, whether in a Council or outside of a Council.
- Although the bishops who are in communion with the head [the Pope] and members [other bishops] of the college, whether individually or joined together in conferences of bishops or in particular councils, do not possess infallibility in teaching, they are authentic teachers and instructors of the faith for the Christian faithful entrusted to their care; the Christian faithful are bound to adhere with religious submission of mind (religioso animi obsequio) to the authentic magisterium of their bishops.
Of individual bishops
Individual bishops' teachings are part of the Ordinary Magisterium (magisterium = teaching office; magister = teacher):
The teaching office of the hierarchy under the Pope, exercised normally, that is, through the regular means of instructing the faithful. These means are all the usual channels of communication, whether written, spoken, or practical. When the ordinary magisterium is also universal, that is, collectively intended for all the faithful, it is also infallible.
If an individual bishop teaches something that isn't merely a restatement of infallible teaching, his teaching is not infallible.
Of all bishops collectively
Even when all the bishops teach something, but they do not intend to define it as infallible teaching binding on the faithful, their teaching is not infallible, although it does possess infallible safety (infallibilis securitas), i.e., there is no danger to the faithful's souls in submitting to it.
cf. what the First Vatican Council's great theologian Cdl. Franzelin, S.J., wrote in [*Tractatus de divina traditione et scriptura*]:
The Holy Apostolic See, to whom the guarding of the Deposit has been committed, and on whom the duty and office of feeding the entire Church, unto the salvation of souls, has been laid, can prescribe theological opinions (or other opinions to the extent that they are connected with theological ones) as to be followed, or proscribe them as not to be followed, not only with the intention of deciding the truth infallibly by definitive sentence, but also without that intention, [but] with the need and the intention of exercising care, either simply or with specified qualifications, for the safety of Catholic doctrine. In this sort of declarations, even though there is not the infallible truth of the doctrine (because, ex hypothesi, there is not the intention of deciding this), but nevertheless, there is infallible safety [infallibilis securitas]. By safety, I mean both objective safety as to the doctrine so declared (either simply or with such and such qualifications), and subjective safety, to the extent that it is safe for all to embrace it, and it is not safe, nor can it be free from the violation of due submission toward the divinely constituted Magisterium, that they should refuse to embrace it.