Does prayer become invalidated by doubting the prostration of forgetfulness after completing the prayer, then prostrating as a precaution, or by repeating the prostration of forgetfulness due to forgetting to cover the feet, which might lead to prostrating three times?

Your prayer is undoubtedly valid, and there was no need for the prostration of forgetfulness, because you are afflicted with waswasah (obsessive whispering), and one who is afflicted with waswasah should turn away from these whispers and ignore them. Your performance of these prostrations does not affect the validity of the prayer, and doubt after completing an act of worship has no effect on its validity. Our advice to you is to turn away from waswasah and ignore it, for indulging in it leads to great evil.