1) Question comes who are real muslims?
There is no such thing as Islam any more but Islam(s). There are many different sects. There are differing opinions amongst these sects. Allah knows who the "authentic" Muslims are and He will judge all according to His Mercy and Will.
2) Who really follows Islam?
Allah can only answer this question. Personally I believe that those who sincerely believe in Allah, the Prophets He sent (culminating in Prophet Muhammad), the Angels , the Jins, and the last day of judgment are Muslims. They are followers of the message as revealed to Muhammad in the form of the Quran.
3) Why there is so much trouble in islamic world?
This is like asking why is there so much trouble in the "world" period. It isn't just the Muslim world. You have a very narrow perception of the world if you are looking at these issues in this light. Look at non-Muslim countries in Africa and Latin America, and Asia, which have equally disturbing political regimes in power, abuses of human rights, and reactionary groups in the form of militants and supporters of terrorism.
4) Is it necessary to hijack airplane to save islam?
Save Islam? This is a ridiculous and insulting position. Islam doesn't need to be "saved", it is safe from all harm - Allah has protected His message and this is evident in the fact that the Qur'an remains unaltered in its pristine state, since it was revealed by Angel Gabriel to Muhammad. The plane hijacking was a political decision, not religious. Don't demonize Islam, especially with the limited knowledge you possess.
5) How islamic faith established? 6) Is the purpose accomplished? If yes then How? if no,then why not?
This question begs for a 1000 volume book. You can read about how Islam was established. It has existed since the creation of Man and Woman. Its purpose is to supplant love in the hearts of humanity for both one supreme and indvisible God and other human beings. This goal is challenged daily by the followers of Satan - who wish to spread hate, oppression and division. In the pages of the Qur'an we have the key to success, its a matter of tapping into that power and implementing Allah's messege in our daily lives. If we can do that, we on an individual and community level, can come closer to achieving success in our lives both now and in the hereafter.
Che