Do the mentioned actions—such as the mother raising her voice in all prayers to remember the number of rak'ahs, raising her voice with tasbeeh in bowing and prostration, and audibly reciting the Mu'awwidhat and supplicating for her children during prostration and tashahhud—affect the validity of her prayer?

It is Sunnah (definition) for a woman to recite aloud where it is prescribed to do so, and quietly where it is prescribed to do so. Deviating from this does not invalidate the prayer. The prohibition of reciting the Quran in prostration (sujud (definition)) is established by the Prophet's (peace be upon him) saying: "Indeed, I have been forbidden to recite the Quran while bowing (ruku') or prostrating (sujud). As for bowing, glorify therein the Lord, the Mighty and Majestic. As for prostration, strive in supplication therein, for it is most likely that your supplication will be answered." What is prescribed in prostration is glorification (tasbih) and supplication (du'a). Reciting the Quran in prostration is disliked (makruh (definition)) and diminishes the reward of the prayer, but it does not invalidate it.