Is the smell in the bathroom slippers—which might be urine—considered najāsah (impurity), especially if it transfers to the foot after pouring water, and does the nose become impure by touching the smelly slippers, and is the water falling from the face when washing away impurity considered impure, given that the smell recurs?

The default is that the bathroom floor and anything that touches it, such as sandals, are pure, unless you are certain that they have been contaminated with impurity. As long as you are unsure about the smell and are not certain that it is urine, the default is that the sandals are not impure, and by extension, the feet and nose are even more so. Beware of succumbing to obsessive whispers (waswas (definition)).