Is the prayer valid when one re-reads part of Al-Fatiha, doubts the number of prostrations, says "Allahu Akbar" and prostrates an extra prostration, performs the two prostrations of forgetfulness (sujud al-sahw) after the salam, then recites the tashahhud and the Salawat al-Ibrahimiyyah, and then says the salam from the prostrations of forgetfulness?

If someone doubts having recited one or more words of Al-Fatihah after completing it, they should disregard this doubt, and nothing is required of them. As for re-reciting the words about which there was doubt, this does not invalidate the prayer.

Regarding the second question, the inquirer should have performed the second prostration, about which she was doubtful, in the last rak'ah before the taslim, and then performed the prostrations of forgetfulness (sujud as-sahw) after the salutation. However, her performing the taslim while doubting the completion of her prayer invalidates it, unless she suffers from obsessive whisperings (waswasah), in which case she should disregard the doubt and is not required to perform sujud as-sahw. Nevertheless, her performing the prostration about which she was doubtful does not invalidate her prayer. One should be wary of falling into waswasah.