Is the first valid ghusl sufficient if the person performing it doubts its validity and then repeats it mistakenly, and is washing the hands obligatory in a sufficient ghusl?

If a person in a state of major ritual impurity (junub) performs a valid ghusl (ritual bath) and then subsequently doubts its validity, he should not pay attention to this doubt. Doubt concerning an act of worship after its completion does not affect its validity. If he repeats the ghusl, the first ghusl is sufficient for him. A sufficient ghusl requires the flowing of water over the entire outer body, and it includes washing the hands. If he washes once and covers his head and body with water, and he does not perform wudu (ablution), it is sufficient for him, after he rinses his mouth and nose, and he intends by it both ghusl and wudu.