Paul claims to be called to be an apostle in each of the undisputed letters (Rom 1:1, 1 Cor 1:1, 2 Cor 1:1, Gal 1:1) as well as several other letters (Eph 1:1, Col 1:1, 1-2 Tim, Titus). In addition to the headings of these letters, Paul refers to his apostleship in several other contexts. In Rom 11:13 he calls himself the “apostle to the Gentiles” and in 1 Corinthians 9 Paul defends his status as an apostle on a par with Peter or Barnabas. But Paul never claims to be one of the Twelve. With the exception of Matthias, the replacement for Judas, this group were chosen by Jesus before the crucifixion.
In fact, in Galatians 1 Paul emphasizes his commission as an apostle but distinct from the Twelve. An “apostle” is someone who is sent as a representative of another, usually some kind of a group. Most lexicons suggests the English “ambassador, delegate, messenger” for the Greek concept of an apostle. Most scholars now associate the Greek apostolos (ἀπόστολος) with the Hebrew shaliach. A person who was sent as a representative or agent acts on the same authority of the sending group.
For example, when the Jerusalem church sent Barnabas to Antioch, it is possible he was send as a shaliach or apostle of the Jerusalem church (Acts 11:22). He would have acted as their representative on the scene should questions arise. Paul is not an apostle sent by the church of Antioch to the churches of Galatia, nor is he an agent sent out by the Jerusalem church. He never claims to be one of the Twelve Apostles, in fact Galatians 1-2 make it clear he is not part of that particular group. Paul’s claim in Galatians is that is an apostle of Jesus Christ and God the Father.
In 1 Corinthians 15:9 Paul alludes to his status as an apostle in his discussion of the resurrection. Paul was not a follower of Jesus until his encounter in Acts 9. As is well known, he was a persecutor of Jesus’ followers prior to the resurrection appearance of Jesus. Paul claims in in 1 Cor 15 to be an eye-witness to the resurrection, albeit one with different credentials than Peter or James since he did not know Jesus before the resurrection.
This experience was like an “untimely birth” (ESV). This word (ἔκτρωμα) is used for a stillborn child or a miscarriage. Many commentators think this is an insult Paul faced in his ministry, he is not just a “Johnny-come-lately” or someone who is trying to “jump on the band-wagon,” or that he has some spiritual deficiency disqualifying him from being considered a “real apostle.” Rather than responding to an attack, Paul is simply listing himself as the final witness because he was the final witness, and his experience is unique among the apostles. But again, he does not claim to be one of the Twelve; like James, the Lord’s Brother, he is commissioned by the resurrected Jesus to be an apostle, but NOT one of the Twelve.
In 2 Corinthians 11, Paul sarcastically refers to his opponents in Corinth as “super-apostles.” But since this rare word can mean superior, it is possible the opponents considered themselves to be superior to Paul and described themselves as his superiors to the members of the Corinthian church. Some have argued this is a reference to the apostles in Jerusalem, but it seems unlikely Paul would refer the Twelve with this snarky title (like added “so-called” to something to question its authority). More likely the super apostles are Greeks in Corinth who have accepted the Gospel but are now behaving like Greek intellectuals. Like many of the other issues in Corinth, Paul is dealing with a pagan worldview in the church.
By way of summary, there was a group called the Twelve who were apostles, and a few other people who were commissioned by Jesus after the resurrection (James and Paul) and were therefore also considered apostles. There were others who claimed to be apostles, like the super apostles mentioned in 2 Corinthians who claimed authority as apostles but were not commissioned by the resurrected Jesus.
What is Paul claiming when he calls himself an Apostle? What does it mean for a letter like 1 Thessalonians, where he does not use the title but then says he could have made demands as an apostle of Christ?
But perhaps the situation was not as strict as Josephus would have us believe. Gentiles were not totally excluded from Jewish worship. The Temple in the first century was expanded to include a large “court of the Gentiles” providing a place for Gentiles to worship. On a number of occasions in the gospels Jesus speaks with Gentiles, although usually the faith of the Gentile is in contrast to the unfaithfulness of the Jews (see for example 
Second, Paul says he has worked harder, been in prison more, been beaten countless times and has been near death many times. Paul uses a series of adverbs (περισσοτέρως twice, ὑπερβαλλόντως once, and πολλάκις once) to overemphasize his difficult life as a servant of Christ. These were not one-time problems he endured for a short time. This is the constant state of his life since he began his ministry!
In fact, Acts describes Paul as a bitter opponent of the gospel. Paul makes a similar statement in Galatians 1:13. Paul likely began to oppose the preaching of Stephen in the Greek-speaking Synagogue of the Freedmen (Acts 6:1-8:1). As a Hellenistic Jew from the Diaspora Paul have fellowshipped at this synagogue, but as a Pharisee he would have been shocked and offended by Stephen’s claim Jesus was the messiah, God had raised him from the dead, and he was going to return soon to render judgment. (Although we do not have Stephen’s speeches before his final one prior to being stoned, it seems likely he would say the same sorts of things Peter and John did in Acts 2-3.)
Imagine contemporary celebrations like Mardi Gras, New Year’s Eve, or Super Bowl parties. There are all excuses for indulging in behavior that would be inappropriate at other times. But this goes beyond the pale: imagine someone behaving so badly at Mardi Gras that people thought they were crossing the line.