Is it necessary to press down on the prayer rug during prostration if it is elevated, and is my prayer valid if I do not press down and ignore the matter, considering it a whisper (waswasa)?

The usual fluff on prayer rugs does not prevent the completion of prostration. Your prayer is valid, and you do not need to repeat it. It is recommended to consistently make supplication, show humility, ward off obsessive thoughts, and consult a psychiatrist.