Does the NP "the Middle East" contain only the determiner "the" and the proper name (noun) Middle East? Or does it contain the AdjP which is the adjective "Middle"? Or is "the Middle East" a noun (a proper name) on its own? (Since people only say "the Middle East", not "Middle East")
"Middle East" is generally considered a single lexeme. While it started out as an ordinary adjective + noun combination (along with "Near East" and "Far East"), it's "fossilized" now: you can't swap in the *Central East or the *Close East, for example. So syntactically, it's reasonable to treat "Middle East" as a single, unbreakable unit.
"The" is then an ordinary determiner; "the Middle East" is syntactically the same as "the apple" or "the book" (which might be an NP or might be a DP depending on the particular theory you're using). It is indeed a proper noun, but certain proper nouns require definite articles in English: compare "he looked at the sun" against *"he looked at sun", or "she's swimming in the Pacific" against *"she's swimming in Pacific". This is one of them.