What is the ruling on a drop of blood appearing with the intention of menstruation after a full day of the usual menstrual bleeding having stopped, and is it permissible to fast and pray after seeing it?

Yellowish or brownish discharge is not considered menses unless it occurs during the accustomed period or is connected to blood. So, if this brown drop appeared after the end of your usual period, it is not considered menses, and you are not required to perform ghusl. However, if it occurred during your accustomed period, then it is considered menses, and you must perform ghusl before praying.