1. Titus's mission to Corinth was to address an issue that would require disciplinary action (if the Corinthians failed to repent) (see 2 Cor 1:23; 2:2,6; 7:11). The same is true of Timothy's mission (1 Cor 4:21; 5:2,11).
2. There was a long-standing clash of lifestyles between Paul and some (many?) of the Corinthians. Paul, in imitation of Christ, exercised self-control and put the needs of the community first (1 Cor 8:13; 9:25-27). Some of the Corinthians, however, indiscriminately followed slogans like "all things are permitted" (1 Cor 6:12; 10:23) and were complacent about sexual immorality, idolatry, and food sacrificed to idols. Paul urged the Corinthians to imitate his lifestyle (1 Cor 4:16-17; 10:33-11:1).
Now, I suggest that the libertines responded by trying to turn the community against Paul. In the context of this clash of lifestyles rejection of Paul meant rejection of his lifestyle, and zeal for Paul meant zeal for his lifestyle. This explains the otherwise obscure connection between 1 Cor 9:1-3, where Paul addresses challenges to his authority, and the previous verse, 1 Cor 18:13, where he outlines his approach to food sacrificed to idols. Those who claimed the right to eat indiscriminately had tried to discredit Paul because of his opposition to their doctrine.
It also explains the connection between 2 Cor 6:14-7:1, where Paul urges the Corinthians to reject idolatry, and both 2 Cor 6:13 and 2 Cor 7:2 where he asks them to open their hearts to him. In the context of the clash of lifestyles, the implication is that they are to open their hearts to his way of life that rejected idolatry. Further evidence of an ideologically motivated attack on Paul by the libertines is found in 2 Cor 12:21-13:7 where they question whether Christ is speaking in Paul.
Paul had to defend himself against criticisms that were intended to discredit his lifestyle. He did so, not for the sake of his reputation, but for the sake of Corinthians, who needed to imitate his lifestyle. The Corinthians had misunderstood the motivation for his self-defense so he had to explain that it was for their benefit (2 Cor 12:19-21). We should not misunderstand Paul's letters in exactly the same way that the Corinthians had done. See Sean's post for further thoughts on the way Paul defends himself only as a means to bring his hearers to greater Christlikeness.
It also explains the connection between 2 Cor 6:14-7:1, where Paul urges the Corinthians to reject idolatry, and both 2 Cor 6:13 and 2 Cor 7:2 where he asks them to open their hearts to him. In the context of the clash of lifestyles, the implication is that they are to open their hearts to his way of life that rejected idolatry. Further evidence of an ideologically motivated attack on Paul by the libertines is found in 2 Cor 12:21-13:7 where they question whether Christ is speaking in Paul.
Paul had to defend himself against criticisms that were intended to discredit his lifestyle. He did so, not for the sake of his reputation, but for the sake of Corinthians, who needed to imitate his lifestyle. The Corinthians had misunderstood the motivation for his self-defense so he had to explain that it was for their benefit (2 Cor 12:19-21). We should not misunderstand Paul's letters in exactly the same way that the Corinthians had done. See Sean's post for further thoughts on the way Paul defends himself only as a means to bring his hearers to greater Christlikeness.
Let us turn now to the tearful letter. Paul wrote the letter "in order that your zeal for us might be made known to you before God" (2 Cor 7:12). Also, it does seem likely that the offender of 2 Cor 2:5-11 had criticized Paul in some way. The tearful letter and the offense therefore concerned the Corinthians' attitude toward Paul. Now, 1 Cor 4:3 and 2 Cor 13:7 show that Paul was not primarily concerned with what the Corinthians thought of him, and in 2 Cor 2:5,10 he even questions whether an offense had been committed, and in 2 Cor 7:12 he denies that he had written on account of the offender or on his own account. So why all the fuss? How can Paul, who cared little what people thought of him, have required the punishment of someone whose criticism had caused Paul little distress? And how can the Paul of 1 Cor 4:3 and 2 Cor 13:7 have written out of much distress and anguish of heart (2 Cor 2:4) just to make the Corinthians zealous for him? This seems inconsistent and egotistical.
The contradictions are resolved when we realize that the tearful letter was sent to counter the libertine doctrine. This issue of licentiousness would have caused Paul to write "out of much distress and anguish of heart" to bring the Corinthians back to zeal for him, meaning zeal for the lifestyle that he exemplified. In 2 Cor 7:12, as in 2 Cor 6:13; 7:1 and 1 Cor 9:1-3 we can assume that Paul expected his hearers to realize (from their familiarity with the context of their recent interactions with Paul) that Paul's lifestyle is in view. The offender's criticism of Paul was an attempt to turn the community against the imitation of Paul's lifestyle. The criticism, in and of itself, caused Paul no great pain, but it posed a huge danger to the community and therefore warranted punishment. Similarly the boasting in sexual immorality in 1 Cor 5:6-8 is compared to yeast that corrupts the whole dough. Paul's primary concern was not to defend himself against criticism, but rather that the Corinthians avoid the licentiousness that might result from that criticism (2 Cor 13:6-7). In the tearful letter Paul had addressed the criticism of himself but his motive had been misunderstood, for he had to explain that had not done so on account of himself nor on account of the offender, but that the Corinthians might have zeal for his lifestyle. This misunderstanding explains why Paul is cautious to avoid a repeat of the same misunderstanding (2 Cor 12:19-21).
So, we have deduced that Paul wrote the tearful letter to encourage zeal for his lifestyle, probably in opposition to the Corinthians' licentious lifestyle. This is exactly the same purpose for which Paul had sent Timothy to Corinth (1 Cor 4:17). Another parallel is that in both cases there is an indication that the zeal lay dormant and needed only to be re-awakened: in 1 Cor 4:17 Timothy need only remind them of Paul's ways, and in 2 Cor 7:12 he had written so that their zeal might be made known to them. The thoughts are very similar.
3. As I argued previously, Timothy's mission to Corinth was to deal with licentiousness. Here is some further evidence that Titus's mission had the same purpose:
a) For Paul the correct response to licentiousness was to "mourn" (1 Cor 5:2; 2 Cor 12:21), and this was the only issue that made Paul write of his tears (see Phil 3:17-19). So, since Paul wrote the tearful letter "with many tears" (2 Cor 2:4) and commended the Corinthians for having lamented in response to the letter (2 Cor 7:7), we should suspect that it was written to combat licentiousness.
b) It was "impurity, sexual immorality, and licentiousness" that made Paul's second visit to Corinth painful, and this visit was brought to mind by his fear that the same problems would recur on his next visit (2 Cor 12:21-13:2). At the time of Titus's mission Paul had the same recollection of his second visit and concern about his next visit: "So I made up my mind not to make you another painful visit" (2 Cor 2:1). This confirms that Titus's mission was to deal with these same issues of sexual immorality and licentiousness.
c) Paul's intention had been to spare the Corinthians (2 Cor 1:23) so it is doubtful that the tearful letter demanded the punishment of anyone. More likely, the tearful letter called for their repentance and perhaps demanded the punishment of anyone who still remained defiant. In any case, we know that there was one Corinthian who was punished as a direct or indirect result of the tearful letter (2 Cor 2:5-11; 7:11-12). Now, it seems from 2 Cor 2:5-11 that the offender had been shunned by the majority. This is the exact same punishment that Paul demands in 1 Cor 5:11 for anyone who is "sexually immoral or greedy, or is an idolater, reviler, drunkard, or robber".
d) Even allowing for a diplomatic gloss in 2 Cor 7:6-16, it does seem that the tearful letter was successful, except perhaps that a minority did not go along with the punishment of the offender (2 Cor 2:6). The issue that was the primary focus of the tearful letter was therefore one that was largely resolved by the time of 2 Corinthians. The Corinthians' licentiousness was such an issue. In 2 Corinthians we have explicit mention of it only at 2 Cor 12:21-13:7, where Paul seems to warn against a recurrence of earlier sins that some had committed.
So, in conclusion, both Titus and Timothy were sent to Corinth to deal with the licentiousness there. This is further evidence that they were one and the same person. But can there really be any doubt about that?