What is the ruling on prayer and fasting after the third day of menstruation, given that menstruation has become three days of heavy bleeding, followed by days of dryness, then a spot of blood, yellowness, and turbidity, and that the white discharge does not appear until the eleventh or twelfth day?

What you see of blood in the first three days is menses (hayd). What you see after that is a valid period of purity (tuhr), so you should perform ghusl, pray, and fast, and your acts of worship are valid. If the blood returns after that, it is menses as long as it is within its possible time. Any yellowish or brownish discharge connected to it is also menses, so you should abstain from prayer and similar acts, and you will not become pure until the blood and any connected yellowish or brownish discharge ceases.