Is the aforementioned Ghusl correct, and what about washing between the buttocks, the anal sphincter, the nails, and ink on the hands?

The obligatory act in ghusl (definition) for janabah (definition) is to ensure water covers the entire outer surface of the body. This includes washing the anus, the area between the buttocks, and other hidden parts of the body. It is not a prerequisite to wash the anus before commencing ghusl unless there is impurity on it. As for the nails, both their outer and inner surfaces must be washed, and anything that prevents water from reaching the skin must be removed. Slight dirt under the nails is excused due to the hardship involved. As for ink on the limbs, it is not considered a barrier preventing water from reaching the skin, and its removal is not obligatory.