Is the described method for ghusl after menstruation correct and sufficient, especially concerning the intention of purification, the Two Testimonies of Faith (Shahadatayn), the order of washing, and the intention for wudu after touching the private parts, taking into consideration the ruling on touching the private parts and its effect on wudu? Also, is it sufficient to wash the outer ear without the inner part, and do teeth discolorations affect the validity of wudu and ghusl?

The ghusl (ritual bath) you described is valid and sufficient, and it removes both major and minor ritual impurity, as long as you perform wudu' (ablution) after touching your private parts, even if it does not conform to the Sunnah. The prescribed ghusl according to the Sunnah begins with the intention in the heart, then saying "Bismillah" (in the name of Allah), washing the private parts and hands, then performing wudu' for prayer, then washing the head three times, then the rest of the body starting with the right side then the left. If the private parts are touched during the ghusl, wudu' must be repeated. What is obligatory in ghusl is to make the water reach all parts of the body, including the inner parts of the ears, because they are considered external, as stated by An-Nawawi. As for stains on the teeth, they do not affect the validity of the ghusl.