Generally, compound nouns will fall into one of two categories; “open” and “closed.” Closed compound noun examples are, schoolboy, congresswoman, where the compound becomes one word. Open examples where the compound either remains separated or is hyphenated include: vice president, prime minister, jack-in-the-box, jack-of-all-trades.
So if you were writing say a story with fictional characters, they may include Dr Birdbrain, Mr Butterfingers and Professor Highbrow. In this context all these expressions would then be both compound as well as proper nouns! Anglo-Saxon is another compound/proper noun. You can think of many more.
"Gulf of Mexico" is a proper place name noun. (The word gulf as in he bridged the gulf obviously would be a common noun.)
Compounding refers to the joining or combining of two or more words which then function as a single word (either as one word ~ headache, or hyphenated ~ self-government, or as separate words ~ police station) and include expressions such as man-of-war, oil of cloves. Therefore, Gulf of Mexico too would be a compound noun, as would United Kingdom and United States of America. That is, all are proper and compound nouns at the same time.
It is quite common for many words in English to change their part of speech status according to the context in which they are used. This versatile grammatical process is called “flexibility of function,” but this belongs to another discussion!
“Playwright” is a compound common noun but “actor” and “director” are just common nouns (not compound nouns) formed from a verb/noun (act) and verb (direct) through affixation of a suffix.
Compounding nouns (or noun incorporation) where the word-formation is created by joining a noun as the first element with a verb, eg. baby-sit, house-hunt, headhunt, sleep-walk/sleepwalk, actually date back to the Middle Ages in English. One notable example is backbite that was recorded around 1300! ~Dr. Hugh Brennan

I hope this helps.

Eric