Common
The proper noun Caribbean refers to the region of the Caribbean Sea. The word is widely used as an adjective to refer to people of things in or from the region. The noun Caribbean can also refer to a native Carib islander.
The proper noun is Puerto Rico (Caribbean island and territory of the US).
When a proper noun is used as and adjective, it is a proper adjective; for example:Ancient Mexican structures have similarities to ancient Egyptian structures.
The correct spelling of the proper noun is Caribbean (sea north of South America).
They are all nouns, Caribbean being a form of "Caribbean Sea" or "Caribbean region." But Caribbean can also function as a proper adjective (Caribbean tribes, Caribbean music).*The word island is usually considered a noun adjunct (attributive noun) when used with another noun, e.g. island paradise, island map. This also applies to location in location scouting and reunion in reunion committee.
That is the correct spelling of the proper noun Gulf Of Mexico (north of the Caribbean Sea).
In the form that you have provided, mon is not even a word. It's not an abbreviation because it doesn't have a period after the 'n; it's not a proper noun, because the 'm' is not capitalized. To abbreviate the proper noun Monday, it is Mon.
Caribbean
The likely word is the plural proper noun Caribbeans, referring generally to natives or residents of the Caribbean islands.
"Park Avenue" is a proper noun, because it is a place. Proper nouns like this should always be capitalized.
No, the noun 'jewel' is a common noun, a general word for any kind of jewel.A proper noun is the name of a specific person, place, or thing; for example:Jewel (Kilcher), American singer, song writer.Jewel Avenue, Queens, NYMS Jewel of the Seas, Royal Caribbean cruise ship