What conditions give other countries the right to intervene in the United States to protect their economic or political interest?
There are at least two ways in which to answer this.
First, no country ever has the legal right to unilaterally intervene in another countries affairs just for economic or political reasons. Countries may intervene in other countries as part of United Nations sanctions. However, it is not as if China or Russia or any other country ever has a legal right to intervene in the affairs of the US.
Second, you can look at this from a realist point of view. Realists do not believe in legal rights. They believe that the world is an anarchic place and countries should and will intervene in the affairs of others whenever it is in their interests to do so. This is not exactly a right, but it is something that countries will do.
So, no country ever has a legal right to unilaterally intervene in the affairs of the US, but realists would argue that countries should do so if it is in their interest.