1 - Bishops are the original ministers of the Sacrament of Confirmation. Eastern Catholic Priests normatively confer Chrismation when they baptise infants. In the Latin Rite, priests may confirm by delegation or a grant of the law itself which is apart from the action of the bishop. Personally, I have confirmed far more people by the grant of the law itself than by a delegation of the bishop.
2 - The law itself (and not the bishop’s delegation) confers the faculty to confirm in the cases of a priest with legitimate canonical office receiving a person he instructed into full communion with the Roman Church. (This, and not a delegation, is the norm for confirmations that are occurring during the Easter Vigil.) Indeed any presbyter is granted the faculty by the law itself to confirm someone who is in danger of death. And, moreover, a priest who is equivalent in law to a bishop has the habitual faculty to confirm by virtue of his office…as would vicars general.
Can. 883 The following possess the faculty of administering confirmation by the law itself:
1/ within the boundaries of their jurisdiction, those who are equivalent in law to a diocesan bishop;
2/ as regards the person in question, the presbyter who by virtue of office or mandate of the diocesan bishop baptizes one who is no longer an infant or admits one already baptized into the full communion of the Catholic Church;
3/ as regards those who are in danger of death, the pastor or indeed any presbyter.
3 - The sacramental life of the particular church continues while awaiting the nomination of a new bishop. Pastors of parishes confirm candidates of the RCIA by the grant of the law itself, quoted above. In other circumstances where there is no diocesan bishop and there are children to be confirmed, confirmation and its provision is made through the diocesan administrator, pending the nomination of the new bishop. I am not sure why it is written “no sacrament can be conferred” since all aspects of the diocese’s sacramental life assuredly continue while awaiting the nomination of a new bishop.
There are really only two situations that are problematic. If he is himself not a bishop, the diocesan administrator would need to issue a dimissorial letter and arrange with a bishop for conferral of sacred order – deacon or priest – which would be conferred by either a retired bishop or the bishop of another diocese. In such a case, the administrator requires the consent of the diocesan college of consultors to issue a dimissorial letter. And, finally, assistance would be needed (again assuming the administrator is not a bishop) for the Chrism Mass, since the sacred chrism must be consecrated by a bishop and therefore a bishop from somewhere must preside at that particular Mass.
4 - Ordinations can be done by any bishop, provided there is a dimissorial letter from whatever is granting incardination. Auxiliary bishops will not infrequently confer, for example, diaconate. For religious order/congregation priests, many times it is a bishop who belongs to the order/congregation who will confer priesthood. This latter case requires a dimissorial letter from the major superior, typically an abbot or a provincial or their equivalent, according to the governmental structure of the incardinating institute. The Rite of Ordination provides the formula for the solemn promise of obedience when the ordinand is not the subject of the ordaining bishop.
5 - As a professor of theology, I can’t imagine why this is even posited today. There is no circumstance in which a bishop is unavailable to confer priestly ordination either by coming to the candidate or the one who is to be ordained traveling to receive ordination. This was so even in the severest persecutions in the concentration camps of Nazi Germany and behind the iron curtain after World War II. Even the most remote place in the world is still part of a bishops’ conference or structure of governance.