Does the usual menstrual period end at sunset on the sixth day, and if I verify purity by placing a white cloth and no trace of blood appears, but then upon washing, a trace of blood comes out, and then I re-verify and nothing appears, is the purification valid or is it necessary to repeat the ghusl?

If your regular period is six days, then after the six days have passed from the time you first saw blood, your period has ended. If you see blood after the duration of your regular period, it is still considered menstruation, because the regular period can increase or decrease. If you perform ghusl (definition) after the cessation of blood, yellowish discharge, and brownish discharge, then your purification is valid.