Is the prayer valid for one who hastened in bowing (rukūʿ), then was overcome by misgivings about the validity of his bowing, especially if it was the last rakʿah, then intended that it was the Sunnah of Fajr, and completed his prayer ignoring the misgivings, then repeated the prayer in another congregation?

What you did was incorrect. That prayer of yours was not sufficient, neither for the Sunnah of Fajr due to lack of intention for it, nor for the obligatory prayer due to its interruption. However, since you repeated it, you have discharged your responsibility. We advise you to ignore waswas (whispers/obsessive thoughts) and pay no attention to them. If the Imam precedes you in ruku' while you are standing, then bow and attain طمأنينة (tranquility/stillness) and then follow your Imam. His precedence will not harm you.