Is there any state in the world which is a true Muslim state, following only the Laws of Islam? If yes, please mention the name. If no, then how can we tell non-Muslims what our society, culture and laws are and where they can be witnessed?

Your question suggests that Islamic invitation to non-believers assumes an ideal Islamic State to convince its addressees. The truth is that although a good Islamic environment is always helpful in convincing many people, it is not an essential precondition for delivering the message. Islamic message is first and foremost based on its beliefs. There are fundamental questions about life which every human being no matter wherever he/she lives wants to get answers to. The Islamic message addresses those questions more than anything else. However, we do need to have certain norms which are to be followed if the basic presentation of Islam is accepted. Some of those norms are applicable to our individual life and some to our collective life. A non-Muslim is certainly more likely to be attracted towards the Islamic message if there are good examples around, both individual as well as collective. However, the message itself is not dependant upon these examples for its authentication. However, given the fact that we know that God Almighty would take into account the circumstances of an individual while deciding about his/her fate in the Hereafter, in case of those who received the message of Islam in an environment where no good examples were available, He is likely to treat such cases more leniently.

Courtesy http://www.islamicissues.info/qa_question.php?qid=129