Why do we bow down on the floor during the Yom Kippur prayers, (I think we do it several times), which is something we never do during the rest of the year? Is there nothing idolatrous about this?
You are referring to a specific part of the chazan’s repetition of the musaf prayer which recounts the order of the service that the High Priest performed in the Holy Temple throughout the entire day of Yom Kippur. It is therefore called the seder avoda of the Cohen Gadol.
When the Temple stood, the eyes of all Israel were raised towards the Cohen Gadol’s order-of-service which began before the break of dawn on Yom Kippur and lasted until the end of the day. The atonement for all of Israel depended on this avoda, which, when performed properly, was manifest for all eyes to see.
During this service, the Cohen Gadol pronounced the ineffable Name of G‑d ten times. Three times during each of the Cohen’s threefold confession on behalf of: 1] Himself and his household, 2] His fellow cohanim, and 3] For the people. These three, threefold recitations comprise nine times. The tenth time the Cohen Gadol pronounced the Name was as he cast lots on the goats.
When the cohanim and the people heard the glorious and awesome Divine Name pronounced by the Cohen Gadol in holiness and purity, they kneeled, bowed down, and fell on their faces saying, “Blessed be the Name of His glorious majesty unto eternity!”
It is during this part of the recounting of the service of the Cohen Gadol that we bow down, as was done in days of old during Yom Kippur in the Temple. However, nowadays, the actual Name is not pronounced, and we bow only three times – once instead of three times for each of the three recounted confessions.
The ineffable Divine Name is so holy that without G‑d’s help, even the Cohen Gadol on Yom Kippur was incapable of bringing it forth from his mouth. Even angels cannot pronounce it. No tongue contains a word for it, because it is concealed, ethereal, and awesome beyond all else.
According to the Shla in the name of Tzror HaMohr (Shla, vol. 6, Yitro, Torah Or 5, p 245) this is the meaning of the verse, “In every place where I will cause My Name to be pronounced, I will come to you and bless you” (Ex. 20:24). The verse should have stated, “Wherever you will pronounce My Name”. We learn from this that flesh and blood is incapable of pronouncing the ineffable Name, and that it was actually G‑d who was pronouncing it from the mouth of the Cohen Gadol.
In synagogues whose floors are of stone, it is customary to place something like a towel upon which to bow, because of the prohibition of bowing down upon a smooth stone, which is associated with idolatry (Lev. 26:1). Other types of surfaces are generally not considered to be problematic. Also, generally, it is considered enough to interpose between the stone floor and one’s face, so even a paper towel or napkin would serve as a separation for this purpose.