I will start off with the Navarre Bible New Testament Compact Edition Explanatory Notes to the vv. 2:2, 2:3, and 2:7 in the Second Letter of Paul to the Thessalonians and then I will quote a great portion of the note to 2 Thes 2:1-12 from the same Bible, after which I will attempt to answer the 4 questions you have raised.
The Explanatory Notes
2:2: Paul warns against over eagerness to expect the Second Coming,
and specific various signs to be looked for first.
2:3, the man of lawlessness: i.e., the Antichrist.
2:7: Evil will operate secretly till the final unmasking.
The note to 2 Thes 2:1-12
Against those who were misleading the Thessalonians by telling them
that the Second Coming was imminent, St. Paul uses Old Testament
images and symbols to assert that two things must happen first -
"rebellion" or apostasy, and the appearance on the scene of the man of
lawlessness" (v.3). Our LORD himself predicted this rebellion (cf. Mk
13:22); he said that when it had run its course, the End would come,
and the last judgment. We do not know what the Apostle means by the
"man of lawlessness" - maybe all the forces of evil taken together as
a tool at the service of Satan, although the description of this
adversay of God is very like that of the "Antichrist" pf St. John (cf.
1 Jn 2:18-22; cf. CCC 675). [...]
Next, stating the Church Teaching on Endtime events
From CCC 673-677, Colin B. Donovan, STL1 provides the following chronology of the general order of events the End as follows:
the full number of the Gentiles come into the Church
the "full inclusion of the Jews in the Messiah's salvation, in the wake of the full number of the Gentiles" (#2 will follow quickly on, in the wake of, #1)
a final trial of the Church "in the form of a religious deception offering men an apparent solution to their problems at the price of apostasy from the truth." The supreme deception is that of the Antichrist.
Christ's victory over this final unleashing of evil through a cosmic upheaval of this passing world and the Last Judgment.
1. cf. Endtimes, Millennium, Rapture | EWTN.
Q1. Has the "the apostasy" (ἡ ἀποστασία) already come? If so, what is it, and when did it occur?
A1. The Church has not made such a pronouncement and the apostasy is
a final trial of the Church "in the form of a religious deception
offering men an apparent solution to their problems at the price of
apostasy from the truth." The supreme deception is that of the
Q2. If not, what criteria would the Pope and magisterium use to identify the apostasy when it occurs?
A2. In the first place, the Magisterium of the Church is not something apart from the Pope, which can undertake the task of identifying something. It is an authority granted the successors of the apostles by the Holy Spirit, something the Pope and the bishops will use when discerning the truth.
That said, how would the Pope and the bishops identify the apostasy?
From CCC 675 and 676 (sans footnotes) under The Church's ultimate trial
675 Before Christ's second coming the Church must pass through a final trial that will shake the faith of many believers. The
persecution that accompanies her pilgrimage on earth will unveil the
"mystery of iniquity" in the form of a religious deception offering
men an apparent solution to their problems at the price of apostasy
from the truth. The supreme religious deception is that of the
Antichrist, a pseudo-messianism by which man glorifies himself in
place of God and of his Messiah come in the flesh.
676 The Antichrist's deception already begins to take shape in the world every time the claim is made to realize within history that
messianic hope which can only be realized beyond history through the
eschatological judgment. The Church has rejected even modified forms
of this falsification of the kingdom to come under the name of
millenarianism, especially the "intrinsically perverse" political form of a secular messianism.
Q3. Has "the man of sin, the son of perdition" (ὁ ἄνθρωπος τῆς ἁμαρτίας, ὁ υἱὸς τῆς ἀπωλείας) been revealed? If so, who is he, and when was he revealed?
A3. The Church has made no such pronouncement and he will be revealed just before Christ comes.1
Q4. If not, again, what criteria would the Pope and magisterium use to identify this individual when he is revealed?
A4. Other than what has been presented above, I believe none of us are equipped to say what criteria the Pope and the magisterium will use to identify this individual when he is revealed. From above, the deposit of faith has specific various signs to be looked for first. [cf. Prophesies of the Antichrist By Early Church Fathers]. Please note that the EWTN article states at the time it was written:
As Cardinal Ratzinger recently pointed out (in the context of the
message of Fátima), we are not at the end of the world. In fact, the
Second Coming (understood as the physical return of Christ) cannot
occur until the full number of the Gentiles are converted, followed by
1. See also: Who is holding back the “secret power of lawlessness” (2 Th2:7)? | BH