Are there any comics in the DC universe that describe any kind of laws superheroes have to follow when they are interacting with other nations that are not their own? Is there any international body that gives leaders some say in how superheroes operate internationally?
Some later comics about the Justice League International actually answer this. This DC team is literally sponsored by the United Nations, follows laws that make it so they generally only intervene in other sovereign nations if it is a humanitarian crisis/crime against humanity situation, and first appeared in Justice League #1 back in 1987.The team exists in mainstream comics to this day and acts more like a superpowered peacekeeping force for the UN than complete vigilantes.
In the special "The Origin of Superman!" issue released in 1973 (and reprinted multiple times), it was revealed that Superman, at least, is an honorary citizen of every United Nations member state.
This presumably grants him rights to enter each country and practice his profession there without a visa.