(1) 20 rakah taravih is Sunnah Muakkadah, since the Prophet (صلي الله عليه وسلم) used to perform 20 rakah salah (excluding tahajjud) the whole month of Ramazan. (He performed it individually according to the tradition of Hadhrat Ibn Abbas). Then at the time caliphate, Hadhrat Umar ibn Al-Khattab (رضی اللہ عنہ) persuaded all the Sahaba to perform it with Jamat, so the Sahaba offered 20 rakah taravih the whole month of Ramzan. Thus it became ijama (consensus) of all the Sahaba upon its being Sunnah.
(2) The Prophet (صلي الله عليه وسلم) used to offer 8 or 12 rakah as tahajjud, but he offered 20 rakah as taravih.
(3) It is not necessary to recite the entire Holy Quran once i.e. it is not obligatory. There will be no sin if one does not recite the whole Quran, but one will be deprived of a lot of rewards, for there are lots of rewards in reciting or hearing the Holy Quran in salah. Likewise, one will not get reward of jamat of mosque if taravih is offered at home. Taravih will be valid but it is not good to miss jamat with no excuse and offer it at home.
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband