Are proper names of persons (e.g. Emily Post), considered to be compound nouns? Are the proper names of places or things (e.g. the Mississippi River) compound nouns?
No. Whole phrases can be proper nouns, such as "Rage Against the Machine", which is the name of a band. Just treat the whole thing as a noun.

Your second example, however, is wrong. The proper noun is "Mississippi" - one word. The phrase "the Mississippi river" requires a lowercase "r". "Mississippi river" is indeed a compound noun, composed of the proper noun "Mississippi" and the common noun "river".

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Thanks Rommie. Your response was helpful.
> The phrase "the Mississippi river" requires a lowercase "r"

It seems that most people write "Mississippi River" instead of
"Mississippi river". Microsoft Bookshelf also uses the uppercase "R".

Actually, what's the general rule governing the capitalisation of
common nouns in a compound noun?
I don't know. I've mentioned before that capitalization preference seems to be more like fashion than grammar. I can see why people might regard the whole phrase as a proper noun. It similarly seems fairly common to write "the Starship Enterprise" instead of "the starship Enterprise".

If I had to guess, I'd have to say that "river" (and "starship") are NOT part of the proper noun, but that their capitalization is (like other capitalization issues) a matter of preference. I honestly don't think there are any "rules" here, but I'm happy to be corrected.

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what is a compound proper noun?

The twister jerked the cowboy