Are laws governing transactions of individuals and companies that cross international borders public international law?
At least some laws that govern individual or corporate transactions that cross national borders would be classified as public international law.
Public international law is a term used to refer to law that binds sovereign states. These are things like treaties and agreements that are binding on states because they have agreed to be bound. Some of these laws end up affecting transactions between corporations or even transactions that include individuals. For example, NAFTA is a public international law. It is binding on the three countries that have signed it. NAFTA, of course, has major impacts on the terms of transactions that cross the borders of the three North American countries.
Thus, some laws that govern the transactions that you mention would be classified as public international law.