A couple of things that have been said warrant comment.
For example, the East focuses on three persons into one God while the West has always focused on One God who is presant in Three Persons.
This notion of a distinction between the West's "One to Three" focus in Triadology vs. the East's supposed "Three to One" focus is not historically well attested. While it is true that the Latin West has tended toward a "One to Three" approach, and in several cases formalized it as the default approach in particular theological circles, the East doesn't actually have a easily demonstrated record of favoring "Three to One" over "One to Three." If anything, the Cappadocian Fathers with their understanding of the monarchy of the Father and calling the Father alone "autotheos," tend toward a strong "One to Three" approach. I'm not sure you can get more mainstream Greek East concerning Triadology than the Cappadocians.
I've only seen this notion in Roman Catholic sources with the goal of demonstrating a supposed fundamental difference in Eastern and Western approaches between the Roman Church and the Eastern Catholic Churches, which is how you use it here. However, it's a considerable oversimplification that doesn't do justice to the various theological tradition of East or West. I would suggest sidelining such notions in a serious dialogue between the positions of the RCC and the EOC.
Now, the Roman Catholic Church has always accepted that all things come from God the Father alone.
This sentence needs a considerable amount of clarification. The Orthodox believe this on its face. However, a perfectly acceptable teaching within the RCC is that the Holy Spirit receives His nature and substance from both the Father and the Son equally as one single principle (this happens to be the position outlined in the Catechism of the Catholic Church which draws from the Council of Florence in 1438). This is contrary to the Orthodox faith.
Do you mean to say that the person of the Holy Spirit, His personhood and existence (hypostasis) and His nature (being God), derive from both the Father and the Son as one principle as I would read your Church's Catechism to imply? Or do you mean something else?
On occasion, the Filioque has been elaborated upon in a way which says that the Holy Spirit proceeds from the "Being" of the Father as opposed to His person. Is this what you mean? The Orthodox say that the Son and the Spirit are from the person of the Father, that the personhood of the Father is the fount of the Trinity and the fount of Being. Among other reasons, this terminology is used to avoid any notion some abstract "Godhood" (or perhaps "Godhead," both words mean the exact same thing") preceding the persons of the Trinity, a notion of existence and nature preceding person. Such a God is not the God of the Christians; it's the God of Plotinus. This is why the Orthodox Church has been so careful it Her use of language on these topics.
Jesus plays an important role on descending the Holy Spirit as the Father does.
Try to keep your terminology consistent. "Proceed from," "descend," "spiriate," etc. all have complex connotations. If you mean here that Jesus Christ sends the Holy Spirit into/to the Cosmos, and particularly to the Church, then you are Orthodox regarding this topic.
Saint Ambrose of Milan said:
The Holy Spirit, when he proceeds from the Father and the Son, does not separate himself from the Father and does not separate himself from the Son"
You have employed this quote incorrectly. It does not deal with spiriation. In context it's concerning the divinity of the Holy Spirit and the fact of the Holy Spirit's unity with the Father and the Son. Saint Ambrose is making the point that if you are baptized into Jesus Christ, God the Son, you are also baptized into the Holy Spirit, the Holy Spirit is God as Christ is God.
A few paragraphs down you will find this passage in Saint Ambrose's work:
"And if you name the Father, you denote equally His Son and the Spirit of His mouth, if, that is, you apprehend it in your heart. And if you speak of the Spirit, you name also God the Father, from Whom the Spirit proceeds, and the Son, inasmuch as He is also the Spirit of the Son."
This is echoing the same sentiment, however, here Ambrose is using a more formulaic understanding of the Holy Spirit's relationship to the Father and Son. Here he is completely in line with the Greek Fathers and the Councils of Constantinople, both the Ecumenical Council in 431, and the Pan-Orthodox Council in 879.
That Council of Constantinople in 879 is likely the one ArmyMatt alluded to when he said "Church as a whole said no filioque, to include Popes. and then later on the West changed." Most historians seem to agree that this Council (which explicitly condemned adding the Filioque to the Creed) was ratified by Pope John VIII. A fair amount of Roman Catholic historians dispute this, though some do agree that the Pope accepted this Council. All Orthodox historians I have read posit that it was in fact accepted. Regardless if it was accepted (a view that I agree with), it certainly wasn't promulgated for very long in the Western Patriarchate
The reason the Greek Orthodox Church rejects the filioque is because of the fact that the Emperor of the East elected a heretical man as Patriarch, which lead to iconoclasm, arianism, rejecting the Papacy etc in the East, which is why the Orthodox are still in schism to this day.
Well, no. I'm not sure who you are referring to. If you've been reading older Roman Catholic sources, I'm going to take a stab and say you're alluding to St. Photius of Constantinople. He was long after Arianism, iconoclasm was dead in the water during his life and preceded his time by about 150 years, and the Papacy is a completely different historical discussion unless you are trying to make an argument concerning the legitimacy of the addition of "Filioque" to the Creed rather than the theological accuracy of it. I would advise starting with clearly defining the concept, demonstrating that it's a valid concept, then showing where it can be found in the Tradition of the Church. Once that's done, you can try to show that the Canons of the Second Ecumenical Council are inapplicable to the historical circumstances (from the Synod of Toledo to Pope Nicholas I of Rome) of the addition of the Filioque to the Creed.