"First, Scripture teaches God alone is all holy."
Rather it teaches not that God alone is all holy but that God alone is holy. In the liturgy right after "The Holy Things are for the holy" we sing "One is Holy, One is Lord Jesus Christ to the Glory of God the Father. Amen.". for Christ alone amongst men is holy. But that does not mean in the words we are denying the holiness of those whom we have just called holy, why beacuse Christ is holy and those participating in Christ are thereby deemed holy. Does that mean all in the Church are holy, by no means, but inasmuch as the Head is holy so also the body. Likewise all are called to be holy "Be ye holy, for I am holy" bishops amongst them, and thus we presume by use of the title to acknowledge the fact that they are holy by participation in Christ and further in being chosen to shepherd his flock; this does not mean that they all holy but we do not because of the unworthiness of some not presume the holiness of the rest, just as we do not knowing that "all sin and fall short of the Glory of God" presume not to call the Body of Christ holy.
"Second, this title is a lie, because it is untrue. The Patriarch is not all holy. So it seems to be an idle word/title"
The Patriarch may or may not be holy, but Christ is holy and so is the seat of the bishop.
It may be your philosophy but what is this philosophy of yours? Is it above the teachings of the Saints? Is it above even the Church? Even Christ? You are relying on your own mind, your own reasonings, and placing this above the teachings of Church. We are only men, we are not perfect, our reasonings have limits, any philosophy we devise no matter how great is in the end foolishness, even as the Apostle writes "Where then is the wise? Where the disputer of this age? Has God not made foolish the wisdom of this world?". Accept then your own limitations, accept that the multitude of the faithful, the many great saints of the Church of God enlightened by Him, have attained far more the reasoning not being limited to one man but the many, or except rather that God has guided the Church into all truth and that God is above the reasonings of all.
"Fourth, Pope Gregory the Great objected to the title "Universal Patriarch" being applied to a bishop, and he said that this title was a precursor of Antichrist. I see no difference between the titles "Universal Patriarch" and "His All Holiness.""
What St Gregory the Great objected to was the idea of a Universal Patriarch which he saw as someone who was the head of the Church and supreme over the other sees, this is not what was meant by such title, and indeed this was in fact what the Papacy became and still is.
"Fifth, the title "His All Holiness" comes from the worldly royal authority, not from Scripture or the apostles."
Because a word or title does not come from the Scriptures does not been it is invalid. The title Theotokos is not found in the Scriptures yet it is meet and right to so name the Virgin Mary.
"Sixth, Orthodoxy teaches that all bishops are equal. How does the title "His All Holiness" comport with this teaching?"
Indeed they are, and this has a great deal of patristic support, however the Church also saw a need to organize around key cities and for these to serve administrative centres, with the bishops thereof heading the holy synods and being places of apeal and administrative decisons. That does not mean all bishops are not equal. The use of a title (more correctly a form of address) is neither her nor there it is a mere form of address not a dogmatic statement of faith.
"Seventh, Why is it that the Patriarch of Constantinople gets this title, when Constantinople is not even an apostolic see?" Historical reasons. As I have said the use of such does not really matter it is not a statement of faith. Beside most sees are not Apostolic.
"Eighth, It would be sinful to call certain people like Patriarch Athenagoras "His All Holiness," when he was in fact a Freemason. Scripture says "Woe unto them that call evil good, and good evil..." (Isaiah 5:20)." For the reasons I have explained above.
"Ninth, We Orthodox Christians believe in Tradition. Can you prove to me that the title "His all Holiness" is part of the apostolic tradition? Who was the first to use the title "His All Holiness"?" No I can't, because it is not part of Tradition it is a title, not an article of faith, not a teaching of the Church. It came to be used, it may one day cease to be used.
"most divine Lord Patriarch of Alexandria, Judge of the World."?
Sorry, it's not going to happen. That title is absolute blasphemy and idolatry. JESUS CHRIST IS LORD."
The title (correctly Judge of the Universe not the World) is in reference to Alexandria setting the Church calendar, i.e. making judgment concerning dating based on the heavens (the sky). It is not a reference to being Judge over the World.
"I don't know if we have valid grounds to criticize the Protestants, since they are only concerned about these non-biblical and non-apostolic innovations. They simply want to return to the simple and pure Gospel of Christ. They want to defend the honor and glory of God. They view the Orthodox and Roman Catholic churches as being eroded by too much paganism." I don't know where you are getting these ideas about Protestants (which cannot really be refered to so broadly) but look at the end result of Protestantism, i.e. Atheism and the modern world.