How, as Eastern Rite, Does your Churches get through such ideas as Purgatory, Immaculate conception, and infalliability of the pope?
Purgatory means a process of purification that the soul undergoes - it’s better understood than a state than a place. We don’t believe in physical combustion there.
Some Orthodox writers call this “Hell”, to which the soul temporarily goes if in need of purification; others use the image of the Toll Houses. Most people think that both the flames of Hell and the joys of Heaven are simply different responses to the love of God, which the soul deified by His Energies experiences as joy, and which the sinner is tormented by because she is unable to get away from and hide from God’s love (this is Nyssa’s description, I think?). Purgatory would be how the soul experiences the love of God as a “purifying fire” - a soul not completely deified, but choosing salvation nonetheless.
The Immaculate Conception simply refers to the purity and sinlessness of the Theotokos, and states that she was sinless through the grace of God and not through her own
unaided efforts. It is not intended to detract from her human nature, but simply to state that she is “more honorable than the Cherubim, and beyond compare more glorious than the Seraphim, who without spot gave birth to God the Word” as our Liturgy of St. John Chrysostom phrases it, and it really doesn’t say anything more. The West means something different by Original Sin than the East does - for the West it simply means the propensity to sin due to the wounding of our nature. Our Lady was subject to temptation, but did not have an innate propensity towards sin.
The infallibility of the Pope is because and only because he is the “rock” upon which Christ built His Church (Matthew 16:18), and since the Church as a whole is infallible, when the Pope speaks
in the name of the Church (i.e., when his words are taken to be those of the Church - when he speaks “ex cathedra Petri”) the Holy Spirit would not permit him to formally define an error as truth. There have been only two times in history when the Pope did this apart from an ecumenical council - when the dogmas of the Assumption and Immaculate Conception were defined - and these times he only did it to ratify and give the Church’s blessing to popular piety and devotions which were not under dispute. Every other example of papal infallibility would be when the Pope gives his seal to the decisions of ecumenical councils - “Peter has spoken through Leo”, as the Council of Ephesus spontaneously proclaimed after Leo formally declared the Council’s work to be proclaimed.
In other words, the Pope isn’t speaking on his own. His voice is the voice of the Church, and when he does speak privately - in his homilies or writings, for example - he may very well make an error, and Catholics will often vociferously disagree with him in such cases without feeling the slightest bit like they are committing heresy.
This is how St. Maximos the Confessor understood what the West calls “Papal infallibility” (sorry, but I’m more versed in the Greek Fathers than the Oriental ones):
“The extremities of the earth, and everyone in every part of it who purely and rightly confess the Lord, look directly towards the Most Holy Roman Church and her confession and faith, as to a sun of unfailing light awaiting from her the brilliant radiance of the sacred dogmas of our Fathers, according to that which the inspired and holy Councils have stainlessly and piously decreed. For, from the descent of the Incarnate Word amongst us, all the churches in every part of the world have held the greatest Church alone to be their base and foundation, seeing that, according to the promise of Christ Our Savior, the gates of hell will never prevail against her, that she has the keys of the orthodox confession and right faith in Him, that she opens the true and exclusive religion to such men as approach with piety, and she shuts up and locks every heretical mouth which speaks against the Most High.” (Maximus, Opuscula theologica et polemica, Migne, Patr. Graec. vol. 90)
) make statements about Orthodoxy which are simply wrong.