Is the 'Umrah valid, and are the prayers valid, and are the black drops menstrual blood or from the effect of medication, knowing that she resumed the 'Umrah in full after those drops stopped descending?

If the aforementioned blood droplets were black and descended at the usual time of menstruation, and their flow did not decrease to less than one day and one night, then it is menstrual blood, and prayer is not valid with it. If she performed Umrah (definition) before purification, her circumambulation (Tawaf (definition)) is not valid according to the majority of scholars. In this case, she must return to Mecca to complete her Umrah, or perform the tahallul al-muhsar (release of one who is prevented from completing Hajj/Umrah) if returning is difficult, unless she stipulated at the time of Ihram (definition) that her place of release is wherever she is prevented, in which case she releases herself by intention.

Some jurists are of the opinion that purity for Tawaf is an obligation but not a condition for its validity; according to this view, the Tawaf of a menstruating woman is valid. The view of the majority is that Tawaf is invalid for one who circumambulates while in a state of ritual impurity. The Hanafis have issued a fatwa (definition) permitting the validity of Tawaf while obligating its repetition as long as one is in Mecca, otherwise, a fidyah (definition) (expiatory sacrifice) is required. There is no harm in adopting this view (i.e., the view of the Hanafis), even though the first view is the preferred one for fatwa.