In an international airport, is there specific set of laws for the international area?

You may have passed a border check, but that doesn't mean you've left the country (just as at land borders: there's a gap between the border posts, but that land is still in one country or the other). So the laws of the country apply. Of course, there are usually some special laws or exceptions that apply to such areas, with respect to immigration and taxes.

The accepted answer provides many useful details but fails to make a very basic point: The laws of the country you are in do apply fully. Except in some very specific cases (e.g. there is a treaty between France, Switzerland and possibly Germany regarding Basel airport), an airport is entirely and unambiguously under the jurisdiction of the country where it's located and therefore subject to its legal system (including EU law in the EU).

Now, some special rules (e.g. regarding taxes and customs duties, immigration, search and seizure, etc.) might apply but those derive from the laws of the country in question, not from some sort of over-arching international rules that would supersede local law.

For example, you might be able to transit without a visa if you don't leave the international transit area, thus being exempted from the regular entry requirement. But the exemption itself results from local law, it's because the airport is in a given country that this rule exists, not in spite of it.

Is there special status with flights between countries with special immigration agreements (e.g. Shengen area)?

Let me answer this first: If you fly from one Schengen country to another, you won't pass a passport control. So the so called "Schengen area" at airports, where intra-Schengen flights depart (not to confuse with "The Schengen area") is not "international" in any way.

But even "international area" at airports is not "international" in the sense of "lawless". Neither will dissidents be able to safely travel through international airports of their home country, for example.

There are special agreements for international air traffic, most notably the Chicago Convention, but AFAIK they do not include the right to transit at every international airport in the world at your will.

Most notably, if you fly for example KLM from Edinburgh via Amsterdam to Chicago, Dutch authorities can and will call you out in Schiphol "international area" if they have any open speeding tickets on your name in their system. Unless you come and pay upfront, you and your luggage won't be allowed to continue the journey.