Does passing wind from the anus affect ritual purity?
Ablution is nullified by the passing of wind from the anus, but washing the anus is not required as wind leaves no impurity.
Should nail polish be removed before making ablution?
Yes, it should be removed because it prevents water from reaching the nails.
What is the ruling on water present on the floor of a toilet/bathroom?
The default ruling is the purity of this water present on the bathroom floor. If one is certain or strongly suspects its impurity, then one washes whatever part of the body or clothing this impure water has touched. If one doubts its impurity, the default is purity, and we do not rule it impure based on mere doubt. And Allah the Almighty knows best.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.