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 is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
The Sunnah in the prayer is that those who are performing prayers should straighten their rows so that every one should join his foot and shoulder to the foot and shoulder of the one who is besides him and they should fill the gaps between them. Prophet said: "Make your rows straight for the straightening of the rows is part of the completion of the prayer."
Anas said: "I saw that everyone of us used to join his shoulder to the shoulder of his companion and his foot to the foot of his companion." [Al-Bukhari]
Ibn 'Umar reported from the Prophet that he said: "Straighten your rows and put your shoulders together and fill in the gaps and be gentle with each other." [Abu Daawood]
Islamic legislation recommends straightening the rows and filling the gaps between people. It does not matter where the feet are placed whether on the rug, off the rug, or partially on the rug, etc.; the important matter is to make sure that the rows are straight.
Spreading a rug on the carpet of the mosque does not mean that it is not clean or impure. Basically the carpet is considered pure, so if a praying person puts his feet or forehead on it his prayer is valid unless he knows that there is some impurity on it.
As for the saying of the questioner "placing of the feet on this clean surface..." we have mentioned that the carpet of the mosques is pure, spreading a rug on it does not mean that it is impure.
However, some scholars are of the opinion that it is preferable to put the forehead on the bare ground during the prostration without any barrier as the Prophet used to do in his mosque. His mosque was floored with pebbles. It is also proved in Sunnah that performing prayers on a mat is allowed as narrated from Anas that the Prophet led his family in the prayer on a mat in his house.
In short all the practices are allowed, whether the person stands on the carpet and prostrates on the rug or stands on the rug and prostrates on the carpet. It is important to straighten the rows so that no one person is ahead of anyone else praying in that row with the excuse of having to stand on the carpet or on the ground.
Allaah Knows best.