Two things should be noted about the following quotation from St. Cyril of Alexandria:
"Since the Holy Spirit when he is in us effects our being conformed to God, and he actually proceeds from the Father and Son, it is abundantly clear that he is of the divine essence, in it in essence and proceeding from it" (Treasury of the Holy Trinity, thesis 34 [A.D. 424]).
First, this text does not speak about the Holy Spirit's eternal hypostatic origin from the Father; instead, it speaks about His work of conforming the Christian faithful to God (i.e., an economic reality); and second, the text can also be translated from the Greek in the following manner: "Since the Holy Spirit when He is in us effects our being conformed to God, and He actually goes forth of the Father and the Son, it is manifest that He is of the divine essence, essentially in it, and of it coming forth." This translation has the added benefit of preventing confusion by avoiding the Latin tendency to equate all the various Greek terms that speak about the Spirit's "movement" or "sending" from the Father through the Son, with His procession (ἐκπόρευσις) of origin, which comes only from the Father.
Pope St. Cyril of Alexendria explicitly taught that the Holy Spirit proceeds (proienai) from the Father and the Son (btw, the Latin original of the line in question of the Athanasian Creed uses the word procedens - if you have not made the connection yet, the Latin procedit is actually a more equivalent translation of the Greek proienai intead of the Greek ekporeusai).
No, procedere is a term with meanings which can accommodate both εκπορευω and προειμι, and that is precisely where the Latins ran into trouble, because they later came to misinterpret passages in Latin translation to refer to the causal origination of the Spirit when in the original Greek they only refer to the Spirit's manifestation through the Son.
St. Athanasius also taught about the relationship between the Holy Spirit and the Son:
"All other things partake of the Spirit, but He, according to you, of what is He partaker? Of the Spirit? Nay, rather the Spirit Himself takes from the Son...When the Father says, "This is my beloved Son," and when the Son says that God is His own Father, it follows that what is partaken is not external, but from the essence of the Father."
(Discourse 1 Against the Arians, Ch. V)
You took that quote out of context. To give a bit more of the immediate context, read this fuller quote, "He, of course, as well as others, must be called Son and God and Wisdom only by participation; for thus all other creatures consist, and by sanctification are glorified. You have to tell us then, of what He is partaker. All other things partake of the Spirit, but He, according to you, of what is He partaker? Of the Spirit? Nay, rather the Spirit Himself takes from the Son, as He Himself says; and it is not reasonable to say that the latter is sanctified by the former. Therefore it is the Father that He partakes; for this only remains to say. But this, which is participated, what is it or whence ? If it be something external provided by the Father, He will not now be partaker of the Father, but of what is external to Him; and no longer will He be even second after the Father, since He has before Him this other; nor can He be called Son of the Father, but of that, as partaking which He has been called Son and God. And if this be unseemly and irreligious, when the Father says, 'This is My Beloved Son [Matthew 3:17],' and when the Son says that God is His own Father, it follows that what is partaken is not external, but from the essence of the Father." http://www.newadvent.org/fathers/28161.htm
St. Athanasius here is speaking of sanctification, which itself is an energy. His argument follows this form: if the Son were a creature, then it should be that he participates in sanctification by the operation of the Holy Spirit, but since it is said that the Spirit receives of the Son in the Scriptures, it is absurd to think that the Son is sanctified by the Spirit, but rather, it must be that the Son is holy by participation in the Father. He must partake of the Father and not something external of the Father, and he therefore partakes of the Father's essence. But, unlike what you claim, St. Athanasius is not teaching that the Son transmits the divine nature to the Holy Spirit. Rather he is speaking of the ordering of the trinitarian hypostases, and how all operations of the Godhead (the example here being sanctification) have their cause from the father, are prepared by the Son, and are perfected in the Holy Spirit.
In his Epistles 56 and 61, St. Athanasius also asserts that the Arian's heresy against the Son is also a blasphemy of the Holy Spirit. Why? Because if the Son is not Divine, then Divinity would not have been transmitted to the Spirit, and the Spirit would be a creature.
I have read both of those epistles, and he does not teach that the Son transmits divinity to the Spirit in either one.
Keep in mind that the term procedit/procedens (the best translation of St. Cyril's proienai) seeks only to express the "transmission" of the Essence of Divinity, and does not seek to denote ontological origination.
There is no such thing as a transmission of the divine nature, as if it were some material thing which could be partitioned. The divine nature is simple, and can only be shared in a sort of natural communion between the Father, the Son, and the Holy Spirit, and the Father alone is the cause of this natural communion and sharing of the Divine Nature (i.e., the monarchy of the Father). The Son himself is not the cause of the Spirit's sharing in the divine nature, despite existing in a mediate position between the Father and the Spirit according to operation, which is why the Eastern Fathers who confessed that the Spirit proceeds through the Son were always careful to add that the Spirit's proceeding through the Son in no way severs the Spirit's natural relation to the Father, because it is from the Father (as the sole Cause) alone that the Spirit shares in the divine nature.