I'm European Union citizen. I have a flight to other continent, with a change in other European country, Schiphol Amsterdam. Am I allowed to leave the "international zone" and go out of airport and then come in again? I'm travelling only with hand luggage. I once heard a technical term for such and extra fee by airplane company. But really, does anyone care and do they let me in again?
Assuming you are an EU citizen, you are indeed allowed to exit the airport and be able to return without problems, as long as you have a valid boarding pass for your next flight. In fact, the Schiphol website suggests that you leave the airport and stroll around the city if you have more than four hours before your next flight. The recommended check-in time is at the latest two hours before flying to another European country, and at least three when flying outside Europe.
Schiphol is one of the largest and busiest airports in Europe, and long waiting times (30+ minutes) for security checks are not unusual.
Are you sure that the first leg of your flight leaves you in international zone of Schipol?
As far as I remember, you still are in European zone, where you can take the train to Amsterdam. International controls take place between the two flights.
In some countries (though not in Europe), there is a tax (normally required to be paid in cash) on either entry or exit to the country. If you're in international transit, you'd be able to avoid that, but entering the landside of the airport and leaving would trigger it. That could be what you're thinking of?
Secondly, there are various flight and airport taxes and charges that the airline collects from you. The situation with a connecting flight is different from two flights. In the UK for example, if your stopover between flights is over 24 hours, you'll have to pay Air Passenger Duty for the 2nd flight, which you wouldn't have to if you're on a short connection. Not an issue on a short break like that, but it could be if you decided to take a longer break.
Oh, and you also need appropriate immigration status to enter the landside. Again for you in Europe as a European, that's not an issue. Changing planes in Dubai and wanting to go landside briefly it would be, and I think we had a question on that very thing quite recently.
Yes you can and there won't be any fee. You just need the necessary documents to enter the Netherlands. But that should be no problem in your case. Be sure to be back to the airport on time.
Schiphol even advertises with the opportunity. They offer a so called Floating Dutchman, also for transfer passengers.