Is what this woman did correct in considering one of the two periods (the one that comes to her for a week and then subsides for a week throughout 3 or 4 months) as menstruation, and the other as istihadah (non-menstrual bleeding), and praying and fasting during istihadah, and stopping for one week in the month?

The first blood that appeared five days before Ramadan is menstrual blood. As for the second blood, not all of it is menstrual blood, because if the duration of the first blood flow were combined with the period of purity in between, it would exceed the maximum duration of menstruation (15 days), making the total 18 days. Therefore, the woman would be in a state of istihadah (non-menstrual bleeding), and she should refer to discrimination (tamyiz). What she discerns as menstrual blood is considered menstruation.

As for the intermittent bleeding that exceeded 15 days and continued without a period of purity less than 15 days, you should also refer to discrimination. What has the characteristics of menstrual blood is considered menstruation, and what does not, is not. Some scholars have held the opinion of referring to the days of her usual cycle, and both opinions are valid. You must make up the prayers missed during the days when the blood was considered irregular (fasad) but you had deemed it menstrual blood.