1- There is nothing wrong with praying on a prayer mat, regardless of whether it is made of rags, palm leaves or anything else, whether it is small or big, like a mat or carpet. 2- It is proven that the Prophet (peace and blessings of Allah be upon him) prayed on mats, carpets and animal pelts.

It is better to recite Du`a from memory during prayer, and the Du`a should be short. If you read Du`a from a piece of paper during the Tashahhud or between the two prostrations, there is nothing wrong with that.