Countries which do not have extradition treaties but do maintain diplomatic relations with the US are Afghanistan, Algeria, Armenia, Bahrain, Bangladesh, Bophuthatswana, Brunei, Burkina Faso, Burundi, Cameroon, Cape Verde, Central African Republic, Chad, China (People’s Republic of China), Ciskei, The Comors, Cote d’ Ivoire, Djibouti ,Equatorial Guinea, Ethiopia, Gabon, Guinea, Guinea-Bissau, Indonesia, Jordan, Korea (South), Kuwait, Laos, Lebanon, Madagascar, Mali, Marshall Islands, Mauritania, Micronesia, Maldova, Mongolia, Mozambique, Myanmar, Namibia, Nepal, Niger, Oman, Principe and San Tome, Qatar, Russian Federation, Rwanda, Saudi Arabia, Senegal, Sudan, Syria, Togo, Tunisia, Uganda, Western Samoa, Yemen, Zaire, and Zimbabwe.

Countries which have neither diplomatic relations nor extradition treaties with the US are Andorra, Angola, Bantu Homelands, Bhutan, Bosnia, Cambodia, Ciskei, Cuba, Iran, Korea (North), Libya, Maldives, Serbia, Somalia, Taiwan, Transkei, Vanuatu, and Vietnam.

If you had to pick one from each section, which would it be?  (with only the knowledge you have now)

Personally, it would probably be China and Vietnam.