Did the mother err by raising her voice in tahlil (saying "La ilaha illa Allah" - There is no god but Allah) and salawat (invoking blessings upon the Prophet) inside the Messenger's Mosque, and what resulted from it, namely the repetition by those present?

What your mother did when entering the Prophet's Mosque, raising her voice, is impermissible, as it is not mentioned in the Sharia (definition). The sacredness of the Prophet (peace be upon him) is the same whether he is alive or dead, and voices should not be raised in his presence, as Allah Almighty says: (O you who have believed, do not raise your voices above the voice of the Prophet). It is more befitting for her to engage in supplication and the prescribed dhikr upon entering the mosque, with tranquility, dignity, and a lowered voice. We hope that Allah will not hold her accountable for what she did due to her ignorance of the ruling.