With a valid visa, is it possible to carry a car for the travel trip in Shengen region? How easy or difficult it is apart of the cost involved? Should owner have the European driving license to drive his car?
Schengen applies to persons, not to objects. For your car the rules for temporary importation in the customs territory of the European Union apply. The customs territory of the EU is not the same as the Schengen area. Norway and Switzerland for example are part of the Schengen area, but not of the customs union. The UK is part of the customs union, but is not in Schengen. You can import a car and use it for up to six months. After that import duties will apply and you will might to re-register the vehicle.
This is a UK page, but the temporary importation rules are the same for the whole of the EU. What might be different in some countries is the time after which you need to reregister a car.
Is it about bringing your own car? Only yesterday I was driving behind a car with Russian license plates not to speak of the many trucks with the most diverse types of license plates. I even have seen a truck with licenseplates from Kuwait, don't ask me how it got here.
If it is about renting or buying a car, different options are available Peugeot even has a formula that caters this situation.
Whether or not you drivers license is accepted does not so much depends on the car you are driving, but more on the regulations between your country and the Schengen area. It is always advisable to cary an IDP.