What would happen, if some 3rd world country declared war to the US?
Nothing would happen. Unless the country had some direct link to Americas security there would be no reason for America to do anything. The period after the WW2 is nothing like today but even then America acted in its national interest and not because of some benevolent mind set. America supported Europe and Japan to prevent the Soviet Union from gaining a foothold. Given the degree of animosity towards the Japanese by America and the other allies powers, had the Soviet Union not been considered a threat, most would have been happy to have seen Japan suffer. It was a strategic imperative that drove America to initiate the Marshall Plan and help stabilise Japan under MacArthur. Hard to believe now but America for a time in the 80s viewed Japan as a strategic threat in terms of trade. Tim Clancy wrote a novel to that effect. America will always act in its own self interest and therefore would do nothing unless a country somehow could present a tangible threat.