Is a prayer valid if performed on a pure prayer rug laid over a mattress that has become impure due to saliva, and is this considered reprehensible (makruh) or an innovation (bid'ah) if the intention is to ascertain and cover the impurity, and not out of arrogance, as stated by Ibn Taymiyyah?

Saliva is pure, so there is no harm in praying on a carpet that has been touched by saliva. There is no need to lay a prayer rug over it. What Sheikh Al-Islam said about placing a prayer rug to avoid doubt regarding impurity is an exaggeration. As for placing a prayer rug to avoid the dirtiness of the carpet (not its impurity), his words do not address that.