The Church of the East is the Assyrian Church of the East. Nestorian is an incendiary term used to refer to a heresy that the Assyrians do not hold, much the same way “Monophysite” is applied to the Syriac Church.
The Oriental Communion includes the Syriacs (and their constituents, the Malankara), the Copts, the Armenians, the Ethiopians and the Eritreans.
Anyway, no, neither communion accepts universalism or annihilationism (in fact I’m pretty sure Ephrem implies in one of his works, perhaps the Hymns on Paradise, that God not destroying the souls in the fire is an act of mercy). Some try to suggest that one of our saints’, Mor Ishaq (St. Isaac), writings suggest universalism but I’ve never been convinced. The Syriac teaching on Heaven and Hell is pretty standard; the select of God go to Heaven and those who reject God burn in Gehenna (hell). Previously souls would go to Sheol (“the Pit” - the shadowy post-death nonexistence the Jews believed in) but by Jesus’s death He destroyed the gates of Sheol and made it irrelevant.