What is the ruling on performing dry ablution (tayammum) on a wall or stone?
One of the conditions for tayammum is that it be with pure, clean dust (soil). It is not valid to perform tayammum by striking stones, a wall, sand that has no dust (i.e., that does not produce dust), pebbles alone, or clay, as these are not from the earth or its same category. 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.
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 speaking while relieving oneself?
It is disliked for one relieving himself to speak while doing so, except out of necessity. And Allah the Almighty knows best.