How is the international trade related to the standard of living of the US as opposed to small industrial nation or of a developing nation?

First of all, most of the smaller nations depend on international trade mainly because of their relatively small domestic economy. By leveraging on larger nations such as US, these smaller nations are able to sell their goods to a bigger market and hence increasing their export revenue and in turn, reducing unemployment. The raise in income will thus lead to higher standard of living. In addition, developing nations focused on low-value manufactured goods or primary products due to their lack of technology expertise. These goods may be labor-intensive as well, for example China.
In comparison, US's standard of living is improved through international trade because the abundance of low-value goods available lowers the international prices and this lower prices benefit consumers. As consumer, you are able to buy more with the same amount of money which is equivalent to higher standard of living.
If US decides to move towards protectionism, what happens is that domestic consumption will rely mainly on domestic production, which is at a higher cost. This prevents the country from reaping the benefits of comparative advantage of international trade.