Is my prayer valid if I pray on a blanket that moves and slips the worshiper when prostrating and placing the hands?

Prayer does not require a prayer rug; it is sufficient to pray on pure ground. Your prostration on a blanket is valid if your forehead is stable on it. However, if it slips and is not stable, your prayer is invalid due to the loss of the pillar of prostration. As for not prostrating on the hands, it does not invalidate the prayer according to many scholars.