How Closely connected was Ethiopia to the Eastern Roman Empire?
Basically everyone agrees that there was substantial contact between Aksum and the Roman Empire in late Antiquity. A lot of attention is paid particularly to the Sixth century, and the Eastern Roman alliance with Aksum against expanding Sasanian power. Especially the question of what role Emperor Justin had in encouraging Aksumite king Kaleb to invade the kingdom of Himyar in what is now Yemen in the 520s.
Whatever the case, it's pretty clear from contemporary written sources, as well as art and numismatic evidence that there was sustained trade and contact between Aksum and the ERE in the Sixth century and into the early Seventh century. At that point, there are the dual dilemmas of Sasanian conquest of Egypt and the later Arab conquest of Egypt which disrupted trade and diplomatic contacts. The second dilemma is that the state of Aksum becomes hazy (kings stop minting coinage, there is a debate whether the city of Aksum is abandoned as a capital).
Everything I have read is silent on the question of diplomatic, religious or trade links between Ethiopia and the ERE after the seventh century.
How closely connected was Ethiopia to Christian community in Egypt and the Levant
Ethiopia was organized into the Diocese of Alexandria, meaning that the Patriarch of Alexandria appointed each new bishop for Aksum.
This meant that there was a continuous, sustained contact between Ethiopia and Alexandria throughout the entire medieval period. However, it is fair to say that there were periodic attempts by Muslim authorities in Egypt to monitor and control that relationship.
Famously in the 1350s, the Mamluk sultan Al-Salih imprisoned Patriarch Marqos. The Patriarch managed to get word to Ethiopian emperor Sayfa Arad, who assembled an army in order to march on Egypt and put pressure on the Sultan to release the Patriarch. The threat worked, and Patriarch Marqos' release was secured.
Even a little earlier, in the reign of Amda Seyon (1314-44), the emperor took an active interest in the upkeep of Christian holy sites in Jerusalem.
How closely connected was Ethiopia to Western European Christians?
This is something that the greatest amount has been written about.
Tadesse Tamrat characterizes the interest of Amda Seyon in supporting holy sites in Jerusalem as part of that emperor's greater interest in the ongoing crusades in the Levant. Tamrat talks about the Dominican monk Guillaume Adam who traveled to Socotra (off the coast of Somalia) in hopes of reaching Ethiopian Christians and cementing an alliance against Mamluk Egypt.
Similarly, David Northrup writes about Ethiopian embassies to Avignon and Rome in 1309 and 1402; Papal letters and a Dominican bishop sent to Ethiopia in 1329 and 1330; and Ethiopian embassy to Aragon in 1429, to Milan in 1452 and to Portugal in 1459.
He also mentions Papal councils in 1437 and 1445 which were intended to address the growing Ottoman threat to Constantinople and to create a united-front among Christendom. Northup mentions that the Patriarch of Alexandria sent two Ethiopian monks stationed in Jerusalem to represent the Ethiopian perspective at that council.
So, there was substantial contact in the Fourteenth and Fifteenth century between Ethiopia and Catholic communities in Southern Europe.
Church and State in Ethiopia 1270-1527 by Tadesse Tamrat, 1972 (2nd printing 2009, Tsehai publishers). Ch 7 "Early Contacts with Christian Europe" pp 248-267.
Africa's Discovery of Europe, 1450-1850 by David Northup. Oxford University Press, 2009. pp 4-5.
Foundations of an African Civilization- Aksum and the Northern Horn 1000 BC - AD 1300 by DW Phillipson. James Currey publishers 2012. This book deals in the archaeology of Aksum, and talks a great deal of Aksumite-Byzantine relations throughout.