Does swallowing residual drops of water after ablution or raindrops invalidate the prayer and the fast? And what should one do during prayer in this situation?

Nasal drops entering the mouth do not invalidate the fast unless any of it intentionally reaches the throat. If it reaches the throat by mistake or out of forgetfulness, the fast is not invalidated. If it is found during prayer, the worshipper should spit it out, and that will not harm their prayer.