le gouvernement américain, le gouvernement des États-Unis
Generally, the French and US governments do get along quite well.
the french monarchy felt threatened
No
The U.S. government was against going into the French Revolution because they had just gotten out of a war, and the government also wanted to remain neutral.
The Korean war
yes
yes
Sounds like you need French legal assistance.
A good place to start would be the French Embassy in the US
There is a prime minister and a president.
The US Government meets at the state capitol for meetings.
The Government chose to stay neutral and not aid with the French Revolution, due to its own internal conflicts such as sectionalism and slavery.