There are a lot of factors in this kind of geopolitics over such a long period of time. This is complicated by the fact that modern Ethiopia covers a territory that between the Ancient period and the formation of the modern state contained a large number of smaller kingdoms, Muslim, Christian, and other, some of which were part of what we might term "Abyssinia," the medieval/early modern predecessor to modern Ethiopia (the Emperor of Ethiopia was styled as King of Kings—ንጉሠ ነገሥት—and some of the heads of local states bore the title of King, ነገሥ). The established chronology that we generally use for Ethiopian history talks about Ethiopian history as if there was a clear continuity between Aksum, Lalibela, medieval and Early modern "Abyssinia", and modern Ethiopia, but the ground reality was much messier and still not fully understood.
It is worth opening up a relief map of Ethiopia to appreciate just how mountainous the highlands are. Aksum, the predecessor kingdom of the Abyssinian/Ethiopian state, was briefly able to project power across the sea to South Arabia (modern Yemen), but during the early expansion of Muslim power, the fleets of Aksum were smashed and it was driven from the islands and the shores. One of the large lowland borders to the northern part of the kingdom is the Danakil Depression, which is famously one of the most inhospitable climates on earth, a hot wasteland full of volcanic activity, with few inhabitants. The entire eastern area of modern Ethiopia (out towards Somalia) was only conquered in the last couple of centuries, the territory of the Emirates of Adal and Ifat (amongst others), which were frequently involved in war with the highlands.It was a major operation to move large armies into the mountains through the country, and even when enemies did, as in the case of the famous invasion of Ahmad Gragn (Ahmad ibn Ibrahim al-Ghazi), who led the forces of the Sultanate of Adal in conquering most of the territories of the highland kingdom, there were still refuges for the Abyssinian forces to regroup and recover (in this case, with a little help from the Portuguese).
During the lifetime of the Prophet Muhammed, some of his family and followers fled to the king of Aksum and were sheltered by them. Accordingly, there is a Hadith, known since the 9th century, which is generally translated as "Leave the Ethiopians alone so long as they leave you alone." This may or may not have had any effect in deterring aggression by Muslim powers: as I have previously noted, the Aksumite fleet was destroyed by the early Muslim state and there were periodic wars between the various Christian kingdoms of the highlands which made up Abyssinia/Ethiopia and their muslim neighbours.
Ethiopia did not survive the colonial expansion of European powers entirely intact—Eritrea's separate existence as a country and a culture is basically a product of the Italian colony that broke apart the otherwise-indistinguishable Tigrinya speakers of northern Ethiopia (Eritrea and the Ethiopian province of Tigray, which has been in the news recently because its ruling party largely instigated and seems to have definitively lost a civil war with the central government). That said, being a Christian power in the area—albeit doctrinally separate and somewhat marginal as a power—definitely gave it a pass from some of the European colonizers, who saw them as potentially an ally or at least benignly ignored them. The Portuguese had a brief but energetic involvement with the Ethiopian state, seeing them as a potential counterbalance to the Muslim powers of the Indian Ocean, which was their sphere of interest at the time. Depending upon the source, 300 Portuguese—or more likely, their guns, picked up by Ethiopian soldiers after they fell—contributed to turning the tide against the otherwise-successful conquest of Ahmed Gragn. The Jesuits were interested in converting the Ethiopians to Catholicism, and succeeded in converting one emperor and instigating a civil war (that the Ethiopian Orthodox Christians won). Several other European powers essentially ignored the Ethiopians—the British apparently had so little interest in them that they did not read letters from the Ethiopian king, only taking notice after he kidnapped a diplomat and then imprisoned the diplomat sent to request the first's release—at which point, the most dominant military power on Earth at the time introduced this fact to the Emperor by marching 25,000 Indian regulars from the sea to the highland fortress of the Emperor, at which point he committed suicide; his successor, from a different dynasty, was largely satisfied with the result and had friendlier, if limited, contact with the British...
The Beta Israel (Ethiopian Jews/Falasha)
As for the presence of Jews in Ethiopia/the Horn, I know this is not the answer you or anyone else wants, but there is essentially no evidence of any continuous jewish presence in the highlands of Ethiopia. On the other hand, there is significant evidence that a group of schismatic Ethiopian Orthodox Christians were labelled as Jews—possibly originally as a term of abuse—and over time took on this identity. That said, their descendants, who mostly live in Israel, are today formally recognized as Jewish by the state.