Do not forget that theopaschism can be plausibly understood in a heretical manner (i.e., by implying Christ suffered in his divine nature - something St. Cyril of Alexandria himself explicitly rejected) which is maybe why so many opposed it at first.
Anyway, the Church of the East denies that it has a Nestorian Christology. You have to be really careful when trying to understand the theology of a church that uses different language, since it may say something that sounds heretical but when properly understood is orthodox. Think of the problems Western Christians had when trying to translate ousia into Latin.
Indeed. The late Patriarch, Mar Dinkha IV, took steps to explicitly disavow hardline Nestorianism. I suspect, within the Church of the East you can find people all across the theological spectrum, with some being closer to the Orthodox, some being heavily Protestant-influenced, and others indeed being Nestorians. Much like Anglicanism (which, like the Assyrian Church, also has open communion and multiple theological "parties" like high- and low-church, etc.)
The Church of the East disavowed what the Imperial Church called "Nestorianism" in antiquity; many of them are puzzled why they are called after a man who was not a member of their Church. The situation is terribly complicated linguistically. The translation of Greek theology into Syriac, and vice versa, is an obvious problem. An oft-overlooked problem is that the relevant Greek terms in Greek
had always been a bit fluid depending on who was using them. And then within Syriac, terms such as "qnomo/qnoma" seem to be used by the East Syrians to mean one thing in Trinitarian theology and another in Christology–this is the origin of the charge that they believed in "two persons" in Christ. Among theologians, there was also the long-running feud between Antiochene and Alexandrian Christologies. And the (strong!) personalities of the various bishops involved in the dispute cannot be ignored, either.
The fundamental disagreement is whether the "communication of idioms" is permissible. "God suffered in the flesh." Is that Orthodox or heretical? Well, it depends on what you mean. If we mean that Jesus Christ suffered (in his passible human nature), and since Jesus is God, we can say that "God suffered in the flesh," then that is Orthodox. If we mean that the divine nature suffered, then that is heretical. Similarly with the statement "God was born of a woman" or any number of statements we want to bring forward. The East Syrian tradition has always been very uncomfortable with the communication of idioms; the other ancient Christian traditions have not had trouble with that.