After ablution, is it necessary to begin by washing the right side three times then the left side three times, or by washing both sides alternately three times? And does ablution remain valid after a ritual bath, or must it be repeated, and likewise when taking a shower without the intention of a ritual bath?

Ghusl (definition) has two forms: a Sunnah (definition) (recommended) form and an obligatory form. Washing the right and left sides three times is recommended, while the obligatory part is to ensure water covers the entire body. Ghusl from janabah (definition) (major ritual impurity) suffices for wudu (minor ritual ablution), even if one did not intend to lift the minor impurity. However, a Sunnah ghusl or ghusl for cleanliness does not suffice for wudu. Whoever performs ghusl without intention must perform wudu afterwards.