The term "Middle East" is actually quite problematic, like many such vague geographic terms. It actually originates as a designation for someplace quite different, what we would now vaguely call South Asia. That Middle East indicated its centrality, between the Far East and the Near East (a term we still use when talking about the ancient world, but much less for the contemporary situation). Once again, like most such geographical terms (e.g. Central Asia, Eastern Europe, West Africa), there are certain core countries that almost always appear in people's lists of the components of the region, in this case, places like Saudia Arabia, Israel, and Iraq. However, some, but not others, would include places like Iran, Turkey, Egypt, and even the Maghreb. I even once had a professor of Ancient Greek try to tell me that Greece was part of the Middle East (of course, I had called it part of Eastern Europe). You may be more familiar with languages that have language authorities, who give their imprimatur to the definitions of words and to grammatical forms. English has no such authority. Consequently, we must only look to usage. Now, I would agree with you that I have rarely seen the term Middle East used without Egypt (except in the old South Asian sense), but there is no authority we can look to in English to establish that such a usage is "incorrect" without some other designation of the context (e.g. within the definition established by some particular organization we are talking about). I could certainly see an organization wanting regional designations to be subdivisions of continents, and so restricting the Middle East to Asia, while assigning Egypt to North Africa, or some similar designation.
The name sounds like a dual construction (like מכנסיים, מספריים), but it's old - this is the name given to the country in the Bible. The duality may or may not be related to lower and upper Egypt being separate entities.
It is also the name of one of Noah's grandsons. Interestingly, the Egyptians themselves did not use a name resembling this. Apparently, the Arabic name comes from the Biblical name.
An interesting thought. Take a look at http://www.abarim-publications.com/Meaning/Mizraim.html#.WxGNne4vyJA.