What should a woman do when she suspects that she stepped on urine with her shoe in the bathroom and then walked with it into the house, and are her prayers invalid? And what should she do about her brother who entered the bathroom with the same shoe and then walked with it into the house? And is it obligatory for her to verify whether she stepped on urine or not?

Things are not judged to be impure merely by doubt, because the default state of things is purity. So, as long as you are not certain that your shoe or your brother's shoe has been contaminated with impurity, it remains pure. The same applies to the house. Your prayers are valid. You must ignore these whispers (waswas) and apply the principle "The default state of things is purity" until the opposite is proven with certainty.