Asian is the proper adjective, and also the demonym (noun) for people in or from Asia.
The adjective form of America is American. The word American can also be a noun (demonym).
Either mongol or mongolian.
The proper adjective for something that originally comes from Oman is "Omani." Oman is an Arab nation in Southeast Asia on the coast of the Arabian Peninsula.
Chinese is the adjective form of the proper noun China. For other uses, 'Sinic' is the word you're looking for, as in "Sinic customs are different from those of Japan". The prefix derived from that is 'sino' as in the phrase "Sino-Indian border".
Middle Eastern is the only proper adjective in the sentence above. It describes "greenhouses."
The proper noun for the proper adjective Taiwanese is Taiwan.
The proper adjective for southwest is "southwestern."
The proper adjective is Haitian.
The proper adjective for Portuguese is "Portuguese."
Vietnamese is the proper adjective for Vietnam.
No, old is not a proper adjective. A proper adjective is an adjective derived from a proper noun, such as Swiss cheese, Bolivian pottery, Chinese silk, etc.
The proper adjective for George is Georgian.