I would like to know the etymology of the word 'sata-gopuram' (the 'chadari' placed on the head) with respect to the prefix 'sata' (or 'chada')
The Sanskrit root word here is “Shata” - शठ. In Vaishnavite tradition, one of Nammaazhvar’s titular names is Shata-kopan. Legend is that after He was born, He drove away Shata-vayu in anger. Keshav Srinivasan has discussed Shata Vayu’s significance above. Since He spurned off Shata in anger, thus the title Shata-Kopa.
As for “chadaari”, the word is actually “Shataari”, which would split as “Shata + Ari” in Sanskrit. The meaning being “enemy (Ari) of Shata”. The Shataaris are always placed under the feet of the presiding deity in sanctum sanctorums of the Vishnu temples, and are made with mini Padukas on top of crown like structures. The belief is that by placing this thing - Shataari - which is always at the feet of the Lord, upon devotee’s head, the devotee is graced with the ability to fight off Shata.