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Though many doubt that St. Paul wrote Hebrews, I believe he did;

The idea that Paul wrote Hebrews has been largely discredited. There are several clues in the book itself. For example Hebrews states:

Heb 13:23 You should know that our brother Timothy has been released, with whom I shall see you if he comes soon.

Heb 2:3 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,

1) Paul would never had called Timothy a "brother" as he looked upon him as a "son". Timothy was also apparently "released" presumably from jail. Timothy had to be incarcerated sometime after Paul's death (sometime around 64-67AD) but before the destruction of the temple in 70AD.

2) Hebrews is an evangelistic call to the Jews. While Paul early on went to the Jews, later he viewed his ministry as to the Gentiles leaving the Jews to Peter.

3) The writer of Hebrews style is very different than that of Paul's letters especially the introduction and the ending.

4) The writer is apparently a "second" generation Christian as the information was "attested" to those who heard. This would be why the writer would have considered Timothy a "brother". He would have been at the same generational line as Timothy.

5) Even the early church fathers Origen and Clement of Alexander had doubts but they went along with the crowd at that time.

There are various theories who MIGHT have written Hebrews but only God knows.

None of the arguments you've advanced to discredit Paul as the author of the letter to the Hebrews are very strong. An argument very difficult that opposes yours, taken from A Consideration Of The Attestation Of Scripture To The Authorship Of Hebrews" (click here)

In a nutshell:

A. Both of Peter's letter are, according to his ministry to "the Circumcision," addressed specifically and narrowly to the Hebrew Christians scattered in the Diaspora, and in the very provinces that were the core of Paul's church-planting efforts.

B. In 2 Peter 3:15 Peter writes:

And account that the longsuffering of our Lord is salvation; even as our beloved brother Paul also according to the wisdom given unto him hath written unto you^plural;

This says that Paul did write to these same Hebrews in the same area.

C. Furthermore, Peter says also that what Paul wrote to them as no just a note, but inspired Scripture according to the wisdom given to him, necessarily by God the Holy Ghost:

As also in all his epistles^{which were later agreed to be canonical}, speaking in them of these things; in which are some things hard to be understood, which they that are unlearned and unstable wrest, as they do also the other scriptures^{OT, of equivalent cananicity}, unto their own destruction.

Paul's other writings were specifically toward particular individuals or particular churches, but Peter equates them as inspired writing.

D. So it may be postulated that: (a) Hebrews is a document that was addressed to the same kind of narrow audience as Peter's was; (b) Paul was a recognized author of inspired Scripture; and (c) therefore Peter is essentially saying that Paul is the author of the Hebrewa letter.

Well, this might be interpreted that the Christian Jews in the mixed churches of Asia Minor were conjointly addressed with the Gentiles in the church by the same epistles, and thus a part of Peter's "you". But what other writer is a Sanhedrin-bent qualified Hebrew scholar and apostle of the first or second water capable of explaining the NT mysteries hidden in the OT but revealed by inspired interpretation in the New?

The Apostle Paul was a spiritual father. However, all Israel are brothers. Paul did call Timothy "ἀδελφός" (adelphos, brother) in four other books, so that argument can be discounted. Paul could very well be the author. The only strong argument was the style of the Greek, which supports the possibility of Apollos.

There is no need to speculate as to who is the author (anonymous Jewish male) as long as one accepts its validity.