This is almost my area, but's it's not (my background and research is fairly firmly rooted in what's going inside the Roman Empire). I'm teaching a course on the first 1000 years of church history and I'm quite keen to include material on Christianity outside the Roman sphere in the first millenium. So I'd be grateful for any material and particularly any recommended resources.
For Africa, you could look at Stuart Munro-Hay's Axum: An African Civilization of Late Antiquity and George Hatke's Axum and Nubia;these resources will also be useful for Christianity in Arabia. Christianity in Arabia is fairly sketchy still; in addition to the material on Axum there's a translation of the Book of the Himayarites that contains some useful materials. The available resources on Christianity in the Iranian sphere are surprisingly extensive, especially if Armenia is grouped within the Iranian sphere. Joel Walker's The Legend of Mar Qardagh is a very interesting primary source on Christianity in the Sassanian empire that also contains excellent introductory material, and the book Holy Women of the Syrian Orient contains an English translation of one of our more valuable sources on Christianity in Iran proper(and surprisingly enough Sassanian Zoroastrianism!), the narrative of the "Martyrdom of Anahid".