Is the prayer valid if the worshiper prostrates for forgetfulness after doubting whether his feet were firmly placed during prostration, or if the worshiper feels gas movement and sounds in the anus during prayer without them being expelled?

What you did is incorrect because enabling the extremities of the feet during prostration is a sunnah (definition) (recommended act), and its omission does not affect the prayer. Returning from a fard (obligatory act) to a sunnah invalidates the prayer if the worshiper does so knowingly and mindfully. However, if they are ignorant, it is not invalidated. Your sensation of gases, if they did not exit, does not invalidate the prayer. But if they exited, such that you smelled or heard a sound, or became aware of the release of gas, then your prayer is invalid and you must repeat it after performing wudu' (ablution).