1. Are Tarawih prayers compulsory or optional? And How many Raka'ah do they consist of?
Tarawih are supererogatory prayers: they are strongly recommended and are Sunnah, not Wajib.
Regarding the number of Raka'ah, the idea that they number only eight is a relatively recent one. Traditionally, Muslims everywhere have offered twenty. This is inherited from the practice of the great caliph, Umar Ibn al-Khattab, during whose time there was no dispute on this issue.
The Prophet (s.A.w.) said, "Follow my Sunnah and the Sunnah of the rightly-guided caliphs after me." And from the period of the Khulafa Rashidun there is a sound Hadith explaining how to perform Tarawih prayers. Umar found people offering Tarawih in different groups in the Prophet's mosque so he gathered them together under one imam, Ubay Ibn Ka'ab.
However, there is room for variation here. Some reports say that at the Prophet's mosque thirty-six Raka'ah were performed. Those who are on the side of eight depend upon a statement of Sayyida Aisha, in which she was talking about the night prayer, Tahajjud of the Prophet and not the Tarawih.
We take the standard which is twenty Raka'ah which is the same number prayed in the Masjid al-Haram in Makkah and in Madinah.
(100 - Solah 7)
2. I have seen some people place the Holy Qur'an on the floor. Is this allowed? It seems disrespectful to me.
Yes, you are right, it is disrespectful. The Holy Qur'an is the word of God. When we are handling the Holy Qur'an, it has to be put in a clean place, no books must be placed on top of it and it is not allowed to be used as a headrest. It is not good to put it on the floor even though the floor may be clean. We have to find a respectable place for it.
(76 - Qur'an 5)