Wouldn't the Orthodox believe the same as the Roman Catholics in this regard; That we know his letters are inspired because so-and-so council established it, or something to that effect?
Sorry to be annoying, since I'm not coming at this from an Orthodox perspective in an Orthodox Forum, but Paul is too interesting not to toss a bit in...
As for my two cents, I don't hold to the whole "inspiration hypothesis" as most people understand it, I believe that the Bible is a useful collection of literature that can help you understand God, but perhaps even more importantly, the nature of reality itself. However, I don't necessarily believe that all of the books have the direct hand of the Holy Spirit on them, nor do I believe that the collection of works commonly referred to as "the Bible" was divinely compiled. Considering that if I'm right, Paul would certainly know it, and believe what he wrote, I'd say he's fine with it. If I'm wrong, his works are divine, sanctioned by every Christian Church in the world, he'd know that, and still be fine with it. Regardless, he'd probably be thrilled to know that his influence on the faith stayed permanent either way, cause who wouldn't be? He's human too, after all, and his writings have essentially made a part of him immortal, as all great poets and writers are. Perhaps even more so, since he, not even being one of the original disciples, set many of the standards by which every church teaches and operates to this day.
I don't know if that tangent I just barfed out is sufficient. I'll go point by point, so you don't have to read it... which I probably should have done from the beginning... I'm leaving the first part in anyway, cause why not?
-Would Paul be horrified to learn that his letters are in the Bible?
Probably not, since he presumably believed everything that he was writing.
-Of all the books in the Bible, the ones written by Paul seem to be most commonly attacked.
Of the New Testament, maybe. I usually here criticism of the Old Testament Law more than anything else, on the basis that if a God could sanction such a primitive guide to society, it shouldn't be considered any being worth engaging with. That's just a nitpick though, sorry for being annoying again.
-Liberals don't like what Paul said about adulterers, fornicators, and homosexuals. Anarchists don't like what he said about paying taxes and being subject to governing authorities.
I consider myself a sort of long-term anarchist (That is, we should strive to raise our children so well that they can't possibly have need of a state). Anarchy in its purest form is simply an absence of compulsory relationships in exchange for an all-encompassing voluntary principle. It's an advocacy for the replacement of the abstracted social contract for a tangible one. It does not refer to the end of rules. Unfortunately, Marxists have corrupted the word to the point where it's essentially a synonym for either unquenchable lawless riot gangs or a Communistic dream-world, neither of which is actually representative of the idea. The word you're looking for here is "post-modernist", not anarchist or liberal. Nothing but semantics really, but maybe it's worth acknowledging.
-How do we know that Paul's writings are God-breathed?
Though I don't know if they actually were, I can think of several good reasons that any Christian could give. 1: They're regarded as infallible and a part of the New Testament, which is the same compilation that includes the entire story of Christ, in every Christian Church on the face of the Earth. 2: They were included in the first complete Christian Bible ever compiled by the Church as canonical writings, which as about as good of an approval and ratification as you can possibly get when it comes to Holy Writings. 3: Even if they hadn't been a part of the Bible, he was still an important founder of the Church, arguably even more important than Simon Peter himself. Can you get more trustworthy?
Hope this helps! I'm running off fumes of my morning energy drink, so if this turned out to be a useless turd of a post, you'll have to forgive me. Thanks!