Is the sufficient ghusl valid if I forgot to rinse my mouth and nose during it, and then performed them with the intention of wudu after the ghusl? And are the prayers performed after that valid?

Rinsing the mouth (al-madhmadah) and sniffing water into the nose (al-istinshaq) are Sunnah acts in ghusl (ritual bath), not obligatory, according to the Shafi‘i and Maliki schools. Based on this, the ghusl and prayer would be valid. However, according to the Hanbali school, al-istinshaq is obligatory in ghusl. Neglecting to perform it consecutively (al-muwalat) does not invalidate the ghusl; rather, it necessitates re-washing the part that was omitted with the intention of ghusl. If you intended to remove impurity absolutely, or to make prayer permissible, your ablution and istinshaq would suffice for the obligation in ghusl. But if you only intended to remove minor impurity, major impurity would not be removed thereby, and it would be necessary to repeat al-istinshaq with the intention of removing major impurity.