You inquired about how the EO and OO view the Assyrian Church. You also wanted to know how the EO view certain OO saints and theologians, since both these churches are in the process of uniting.
First, please read the OO’s view of Nestorius and the Assyrian Church: http://www.britishorthodox.org/nestor.php
Next please read the article about the unfruitful dialogue between the OO and the Assyrian Church. In it is also mentioned the dialogue between the RC and the Assyrian Church which the OO were invited to witness: http://www.britishorthodox.org/assyrian.php
Regarding the dialogue between the EO and the OO, they have issued a declaration on common faith stating that each Church recognized that the other is a true Apostolic Orthodox Church that has maintained the faith. Here is the link of their joint decleration:http://www.britishorthodox.org/2church.php
Note this section that allowed the two Churches to declare that they have the same Orthodox faith:
Throughout our discussions we have found our common ground in the formula of our common Father, St. Cyril of Alexandria : mia physis hypostasis (he mia hypostasis) tou Theou Logou sesarkomene, and in the dictum that "it is sufficient for the confession of our true and irreproachable faith to say and to confess that the Holy Virgin is Theotokos" (Hom : 15, cf. Ep. 39).
Great indeed is the wonderful mystery of the Father, Son and Holy Spirit, one True God, one ousia in three hypostases or three prosopa. Blessed be the Name of the Lord our God, for ever and ever.
Great indeed is also the ineffable mystery of the Incarnation of our Lord Jesus Christ, for us and for our salvation.
The Logos, eternally consubstantial with the Father and the Holy Spirit in His Divinity, has in these last days, become incarnate of the Holy Spirit and Blessed Virgin Mary Theotokos, and thus became man, consubstantial with us in His humanity but without sin. He is true God and true Man at the same time, perfect in His Divinity, perfect in His humanity. Because the one she bore in her womb was at the same time fully God as well as fully human we call the Blessed Virgin Theotokos.
When we speak of the one composite (synthetos) hypostasis of our Lord Jesus Christ, we do not say that in Him a divine hypostasis and a human hypostasis came together. It is that the one eternal hypostasis of the Second Person of the Trinity has assumed our created human nature in that act uniting it with His own uncreated divine nature, to form an inseparably and unconfusedly united real divine-human being, the natures being distinguished from each other in contemplation (theoria) only.
The hypostasis of the Logos before the incarnation, even with His divine nature, is of course not composite. The same hypostasis, as distinct from nature, of the Incarnate Logos, is not composite either. The unique theandric person (prosopon) of Jesus Christ is one eternal hypostasis Who has assumed human nature by the Incarnation. So we call that hypostasis composite, on account of the natures which are united to form one composite unity. It is not the case that our Fathers used physis and hypostasis always interchangeably and confused the one with the other. The term hypostasis can be used to denote both the person as distinct from nature, and also the person with the nature, for a hypostasis never in fact exists without a nature.
It is the same hypostasis of the Second Person of the Trinity, eternally begotten from the Father Who in these last days became a human being and was born of the Blessed Virgin. This is the mystery of the hypostatic union we confess in humble adoration - the real union of the divine with the human, with all the properties and functions of the uncreated divine nature, including natural will and natural energy, inseparably and unconfusedly united with the created human nature with all its properties and functions, including natural will and natural energy. It is the Logos Incarnate Who is the subject of all the willing and acting of Jesus Christ.
Especially important to your question Ben about the Assyrian Church is the following from the joined declaration of faith between the EO and the OO:
We agree in condemning the Nestorian and the Eutychian heresies. We neither separate nor divide the human nature in Christ from His divine nature, nor do we think that the former was absorbed in the latter and thus ceased to exist.
In their second agreed statement (also found in the article I linked to above) the two Churches had this to say about the Nestorian teachings:
2. Both families condemn the Nestorian heresy and the crypto-Nestorianism of Theodoret of Cyrus. They agree that it is not sufficient merely to say that Christ is consubstantial both with His Father and with us, by nature God and by nature man; it is necessary to affirm also that the Logos, Who is by nature God, became by nature Man, by His Incarnation in the fullness of time.
6. Both families agree in rejecting interpretations of Councils which do not fully agree with the Horos of the Third Ecumenical Council and the letter (433) of Cyril of Alexandria to John of Antioch.
Regarding your question as to how the EO will view certain OO theologians and saints that were declared heretics, here is their agreed upon declaration:
9. In the light of our Agreed Statement on Christology as well as of the above common affirmations, we have now clearly understood that both families have always loyally maintained the same authentic Orthodox Christological faith, and the unbroken continuity of the apostolic tradition, though they have used Christological terms in different ways. It is this common faith and continuous loyalty to the Apostolic Tradition that should be the basis for our unity and communion.
10. Both families agree that all the anathemas and condemnations of the past which now divide us should be lifted by the Churches in order that the last obstacle to the full unity and communion of our two families can be removed by the grace and power of God. Both families agree that the lifting of anathemas and condemnations will be consummated on the basis that the Councils and Fathers previously anathematized or condemned are not heretical.
We therefore recommend to our Churches the following practical steps :
A. The Orthodox should lift all anathemas and condemnations against all Oriental Orthodox Councils and Fathers whom they have anathematised or condemned in the past.
B. The Oriental Orthodox should at the same time lift all anathemas and condemnations against all Orthodox Councils and fathers, whom they have anathematised or condemned in the past.
C. The manner in which the anathemas are to be lifted should be decided by the Churches individually.
Again please read their entire joint declaration of faith that details how the two Churches came to these conclusions: http://www.britishorthodox.org/2church.php
Here is also a progress report on the joint commission between the EO and the OO: http://www.britishorthodox.org/progress.php
I hope this answers your questions Ben.