>The reason Jesus’s sacrifice saves us that He lived a perfect sinless life and then received God’s wrath anyway.
No, that idea is called Penal Substitutionary Atonement and it is absolutely not the Orthodox belief about how this works at all.
The Orthodox belief is that Jesus's sacrifice saves us because He was God. A perfectly sinless man or woman could not have replaced Him. The point is that Jesus was both God and man and the same time, which basically allowed Him to exploit a loophole in the rules of the universe to undo the Fall.
After the Fall, all humans who died went to Sheol (Hades), the realm of the dead, the place of separation from God, sometimes improperly called "Hell" (I say "improperly" because this was a different place from the Hell that we might go to now). So Jesus, being a man, also went to Sheol when He died. But He was also God at the same time. So God went to Sheol. God went to the place without God. To use a modern metaphor, you could say this "crashed the operating system of the universe and made it reboot". Sheol was destroyed and all the souls trapped there were released.
St. Athanasius of Alexandria explains it better in his book, On the Incarnation. But the central point is this: We are saved because a man who was also God died. Not just because a sinless man died.
Athanasius On the Incarnation is a classic and it is not very long.
The Catechism quotes one of his most famous lines from this book in [CCC #460] "God became man so that man might become God."
A lot of these questions are answered in the classic 4th century text On the Incarnation, by St. Athanasius of Alexandria. To summarize:
>did Jesus suffer for our sins on the cross?
Yes, in the sense that our sins are the reason why He had to do that. So He suffered and died because of our sins.
>Did he take our place and did so out of love?
No, He did not take our place, we don't believe in penal substitutionary atonement. He chose to become human and die - out of love, yes - so He could descend to Hades/Sheol (the realm of the dead) and destroy its gates and liberate the human souls imprisoned there.
>Was him being crucified significant or could he have died some other way and it have the same effect?
He could have died in any other way and it would have had the same effect, but He arranged events so that He would be crucified, for a number of reasons (these are explained very well in On the Incarnation). Basically, the purpose of the Crucifixion was didactic: Christ died in this way in order to teach us things.
>Was his blood important?
Of course, in the sense that we drink His blood in the Eucharist. We commune through His blood.
But if you mean to ask if it was necessary for Him to die in a way that involved spilling His blood - no, He could have died without spilling blood and it would have had the same effect.
>Are we made clean through his sacrifice?
Yes? No? It depends on what you mean by this.
We are made clean first at our baptism, and then later each time we partake of the Eucharist. These things are made possible by the death and resurrection of Christ.
So, we are made clean through [sacraments made possible by] His sacrifice.
Saint Vladimir Seminary Press has produced some nice little paperback volumes with patristic writings, that go for about $10-$15 each, so it's a pretty affordable way to get them, if you like hard copies rather than reading online. I would particularly recommend their edition of Saint Athanasius "On the Incarnation", both because it's a classic of the patristic age, and because their edition features a preface by C.S. Lewis which is also really excellent.
For your questions on the resurrected body, have a read of The Spiritual Was More Substantial Than the Material for the Ancients.
For your questions on Jesus's death and sacrifice, you definitely want to read On the Incarnation.
Feel free to ask more questions here, but yes, the death of Jesus is understood to rid humankind of the curse of death.
No better place to go than the OG anti-Arian:
I would explain it as the ancient Fathers have:
or the Western Athanasian Creed, sans filioque of course, but that is such a minor point when compared to the drastic differences between Trinitarian Christianity and Mormonism.
Men far wiser than me have written on the topic: St. Athanasius' On the Incarnation Amazon CCEL and St. Basil the Great's On the Holy Spirit Amazon CCEL
More importantly, the LDS's false godhead is not the One Triune God described by Holy Scripture, and so cannot be the same Trinity. The false "god the father" of mormonism has a physical body, is but one of many gods in existence, was once a mere man, is not the ex nihilio creator of all things, the list of heresies and non-Scriptural beliefs goes on and on. By contrast, despite the vast doctrinal differences between the Eastern Orthodox, Roman Catholic and Protestant Christian groups on the planet, we are all in basic agreement on the Trinitarian Nature of God.
This is well-known collection online has a bunch. St. Vladimir's Seminary Press publishes small editions of individual writings, which are usually available on Amazon. St. Athanasius' <em>On the Incarnation</em> is a good one to start with.
I'd reccomend that you ask that question over at r/Orthodoxchristianity.
I'm a pretty recent convert to Orthodoxy and Christanity so you'd get better responses from more well read people than myself.
On the Incarnation by st. Athanasius is one I would reccomend to think about the incarnated Christ, but that isn't necessarily focused on that specific part of his work and ministry.
The best thing that comes to my mind specifically with regards to your question is the Divine Liturgy of st. Basil the Great written for Great and Holy Saturday (The Lord's Sabbath). Here is a portion of it.
I'm not sure there is a kindle version of this translation, unfortunately. Looks like the paperback is $15: Amazon