What is Tasawwuf or Sufism? What is its definition? When did it start? What is its methodology? Are Sufis right or wrong? What is the evidence for this?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.

We cannot answer your question about Sufism in a Fatwa (religious ruling) or an essay. However, we will shed some light on what may be relevant here, and then refer you to what will benefit you, if Allaah wills.

The definition of Sufism:  There is great difference of opinion among Sufis concerning the definition of Sufism, to the extent that some of them have collected about one thousand definitions.  As-Subki reported that Ibn As-Salah mentioned that he read a book in which the author mentioned one thousand opinions about the meaning of Sufism and the Sufis.  Such great variation in defining this term is evidence of the fact that it is not based upon firm foundations. Rather, it is defined according to the taste and personal experience of each individual in achieving his goals in worship. Shaykh Abd-Ar-Rahmaan Al-Wakeel mentioned that individual inclination is the only way to obtain knowledge in Sufism. For example, Ash-Shibly defined Sufism saying, "Sufism is to keep your senses and watch your breath."

Additionally, the questioner may refer to the following Fataawa, (religious rulings) which address some of the Sufi Tareeqahs (orders): 

By reading these related Fatwas, you will know the truth about some Sufi Tareeqahs.

However, it should also be known that Sufi Tareeqahs differ in how close to or far they are from the Sunnah.  They also differ in terms of their aims in the past, as well as the present.

Therefore, they should not be judged as a whole because some of them are Mubtadi‘ah (depending mainly on religious innovations), others are heretics, while some of them are ascetics.

Finally, we advise you to keep away from all these Sufi Tareeqahs, because they adopt many religious innovations and acts that are misguided.  We advise you to fear Allaah The Almighty, and to seek to acquire correct knowledge of the Sharee’ah at the hands of righteous and pious scholars, who have sufficient wisdom and insight and follow the guidance of the Prophet, sallallaahu ‘alayhi wa sallam. He stated that he has left us the Quran and the Sunnah, and whoever adheres to them will never be lost.  [Maalik in Al-Muwatta’ (Al-Albaani: Saheeh)]

Allaah Knows best.