Is a ghusl for janabah considered valid if some limbs were immersed in a vessel during wudu, and then the body was covered with water without rinsing the mouth (madmada) or sniffing water into the nose (istinshaq), knowing that madmada and istinshaq were already performed during the aforementioned wudu? Is it obligatory to make up the prayers performed with this ghusl, and what is the ruling on fasting during this period? Is it obligatory to make up the Fajr prayers in which wudu was performed from the vessel?

The order of washing the body parts is not obligatory for ghusl (definition) (ritual bath). Ghusl is achieved by having the intention and making water reach all parts of the body. So, if the person performing ghusl rinses his mouth and nose, then makes water reach his entire body, his ghusl is valid and sufficient. The invalidation of wudu' (ablution) does not affect the validity of ghusl, because performing wudu' before ghusl is a Sunnah (definition) (recommended act), not an obligation. If one does not perform wudu' at all, his ghusl is valid by scholarly consensus.

There is no need to repeat the prayers performed with the aforementioned ghusl because it is valid, and purity is not a condition for fasting.

Regarding your prayer in the train or at home upon arrival, please refer to Fatwa (definition) No. 114659.

If you are suffering from waswasah (obsessive whispering), we advise you to ignore and discard it.