Did Paul consider himself to be writing "inspired" words? Or did he think it was just another letter? What about the other writers of the NT?Although this is not part of the question, I think it is helpful that we recognize that the Old Testament prophets believed they were inspired. When they use such language as, "The Word of the Lord came...", "Thus says the Lord...", or other similar phrases, they are confessing that the words that they spoke or wrote down were not their own words, but the words of God. They were claiming inspiration.
Likewise, when Moses gave the Law, he understood that these were not just a bunch of rules that he had made up, but that he was speaking and writing the words of God.
So, now, we come to the New Testament, and read the words of Paul;
For I would have you know, brothers, that the gospel that was preached by me is not man's gospel. For I did not receive it from any man, nor was I taught it, but I received it through a revelation of Jesus Christ. (Gal 1:11-12)When we read these words we begin to recognize that Paul is claiming inspiration for himself. He is claiming that his writings are just as inspired as any of the Old Testament prophets writings were.
Next, we see that Paul placed his own teaching on par with the teaching of Christ about divorce;
To the married I give this charge (not I, but the Lord): the wife should not separate from her husband (but if she does, she should remain unmarried or else be reconciled to her husband), and the husband should not divorce his wife. To the rest I say (I, not the Lord) that if any brother has a wife who is an unbeliever, and she consents to live with him, he should not divorce her. (1Co 7:10-12)In verse 10, Paul summarizes Jesus' teaching on divorce. He says, "the Lord says this". Then, beginning in verse twelve, Paul gives his teaching, which is complimentary to Jesus' teaching, concerning divorce. In verse twelve he says, "I say this".
Should we understand Paul to be saying that this is something to be taken as a lesser teaching that of Christ, and therefore that we can ignore Paul's teaching if we like? Or, is he simply stating the fact that Jesus had actually said these words and then separating the quote from Jesus form his own words which he prefaces by saying, "This is what I say"?
I think the latter interpretation is more faithful to the text. If it is, then Paul is claiming to be so inspired that his teachings are equal with Christ's teachings.
Also, concerning Paul's writings, it is evident that Peter thought Paul to be inspired because he says,
And count the patience of our Lord as salvation, just as our beloved brother Paul also wrote to you according to the wisdom given him, as he does in all his letters when he speaks in them of these matters. There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures. (2Pe 3:15-16)These are interesting words from Peter. Basically, Peter compares Paul's letters to the rest of Scripture. Peter tells his readers that there are false teachers who twist Paul's letters to their own destruction, just like they do the rest of the Scriptures. When Peter says, "Scriptures", what is he referring to? He is referring to all the inspired writings of the Bible! Old and New Testaments.
Concerning the other writers of the New Testament, first of all, we have the words of Jesus concerning the inspiration of the Apostles. He says,
But you will receive power when the Holy Spirit has come upon you, and you will be my witnesses in Jerusalem and in all Judea and Samaria, and to the end of the earth." (Act 1:8)So this is Jesus telling His Apostles that they were to be His mouthpieces in all the world. If they believed Jesus, then they also must have believed that they were speaking and writing the words of Christ. They must have known that they were inspired.
The writer of Hebrews seems to claim to be a spokesperson for Christ in the first two verse of Hebrews, although later, in chapter 2, he seems to say that he is not an Apostle,
Long ago, at many times and in many ways, God spoke to our fathers by the prophets, but in these last days he has spoken to us by his Son, whom he appointed the heir of all things, through whom also he created the world....The general tone of James' letter seems to be of someone who is writing with authority, making several commands throughout the letter, with no appeal to outside sources for their validity, however, he makes no claim of inspiration.
Therefore we must pay much closer attention to what we have heard, lest we drift away from it. For since the message declared by angels proved to be reliable and every transgression or disobedience received a just retribution, how shall we escape if we neglect such a great salvation? It was declared at first by the Lord, and it was attested to us by those who heard, while God also bore witness by signs and wonders and various miracles and by gifts of the Holy Spirit distributed according to his will. (Heb 1:1-2, 2:1-4)
Peter claims inspiration rather explicitly,
For we did not follow cleverly devised myths when we made known to you the power and coming of our Lord Jesus Christ, but we were eyewitnesses of his majesty. For when he received honor and glory from God the Father, and the voice was borne to him by the Majestic Glory, "This is my beloved Son, with whom I am well pleased," we ourselves heard this very voice borne from heaven, for we were with him on the holy mountain. And we have something more sure, the prophetic word, to which you will do well to pay attention as to a lamp shining in a dark place, until the day dawns and the morning star rises in your hearts, knowing this first of all, that no prophecy of Scripture comes from someone's own interpretation. For no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit. (2Pe 1:16-21)John claims to be a 'witness' of Christ (when he says "we..testify", 'testify' means to 'bear witness),
That which was from the beginning, which we have heard, which we have seen with our eyes, which we looked upon and have touched with our hands, concerning the word of life-- the life was made manifest, and we have seen it, and testify to it and proclaim to you the eternal life, which was with the Father and was made manifest to us-- that which we have seen and heard we proclaim also to you, so that you too may have fellowship with us; and indeed our fellowship is with the Father and with his Son Jesus Christ. (1Jo 1:1-3)John is even more explicit when he tells us that he heard the voice of Christ telling him to write the things that he saw in a Spirit-induced vision in a book,
I, John, your brother and partner in the tribulation and the kingdom and the patient endurance that are in Jesus, was on the island called Patmos on account of the word of God and the testimony of Jesus. I was in the Spirit on the Lord's day, and I heard behind me a loud voice like a trumpet saying, "Write what you see in a book and send it to the seven churches, to Ephesus and to Smyrna and to Pergamum and to Thyatira and to Sardis and to Philadelphia and to Laodicea." (Rev 1:9-11)It has been theorized that Jude was claiming to be inspired when he said that his mind was changed about the topic of his letter (the theory being that the Holy Spirit moved upon him to write the words that he ended up writing),
Beloved, although I was very eager to write to you about our common salvation, I found it necessary to write appealing to you to contend for the faith that was once for all delivered to the saints. (Jud 1:3)Matthew, as an Apostle, probably understood that his writings were inspired, for the reasons stated above in Jesus' parting words (Acts1:8).
Mark may be claiming inspiration, by the virtue of proclaiming the same Gospel which Jesus preached and bearing witness to Him (Mark 1:14-15), in the opening of his Gospel when he says,
The beginning of the gospel of Jesus Christ, the Son of God. (Mar 1:1)And finally, Luke claims to have perfect knowledge of the story of Christ and His church,
Inasmuch as many have undertaken to compile a narrative of the things that have been accomplished among us, just as those who from the beginning were eyewitnesses and ministers of the word have delivered them to us, it seemed good to me also, having followed all things closely for some time past, to write an orderly account for you, most excellent Theophilus, that you may have certainty concerning the things you have been taught. (Luk 1:1-4)In any case, I think that there is conclusive evidence to point to the fact that most of the writers of the New Testament believed that they were inspired.