Was my action correct when I prayed and made a mistake in the bowing (ruku'), then performed the prostrations of forgetfulness (sujud al-sahw), then said the salutation (taslim), and sought forgiveness once, then doubted that I had not glorified (Allah) in the first prostration of the prostrations of forgetfulness, so I prostrated for it again?

Prostration of forgetfulness (sujud al-sahw) is legislated for repeating the bowing (ruku') due to an error in it, because it is an addition to the actions. As for doubting whether one omitted the glorification (tasbih) in the first prostration of sujud al-sahw, it does not necessitate another prostration. However, performing sujud al-sahw a second time does not invalidate the prayer.