Is my prayer valid if I doubted having performed the second rak'ah during the first tashahhud, then stood up to perform it, then remembered that I had already performed it and made this rak'ah the third, and completed the prayer and performed the prostration of forgetfulness (sujud al-sahw)?

You stood up from the first tashahhud while doubting the number of rak'ahs, and then it became clear to you that you had stood for the third rak'ah not the second. In this situation, there are two opinions regarding the prostration of forgetfulness (sujud al-sahw). The opinion that mandates prostration is stronger in evidence and closer to caution, because the worshipper performed a part of their prayer while uncertain if it was part of it.

If you interrupted the first tashahhud before completing what is sufficient in it, then you are like someone who omitted the first tashahhud. Therefore, your prostration of forgetfulness is in its correct place, because omitting the first tashahhud due to forgetfulness or ignorance necessitates the prostration of forgetfulness. Your prayer is valid, and you are not required to repeat it.