Is the anus considered impure, and does the handkerchief become impure by wiping or touching it, necessitating washing it with water? And does the same question apply to the inner and outer parts of the vulva?

The default state of the private parts is purity, unless there is certain impurity present.

If the front or back private part is wiped with a dry tissue, then the tissue is pure.

If the tissue is touched by moisture from urine or feces, then the spot touched by the moisture becomes impure, and the impurity must be removed.

Discharges coming from the urinary opening are impure, while discharges coming from the birth canal are pure.