Is my sitting for the first Tashahhud, after I had stood up for the third Rak'ah, and had only lifted my knee while my thigh was still touching my leg and heel, and I did not perform the prostration of forgetfulness (Sujud al-Sahw), valid?

Your return to the Tashahhud before you had fully stood up was correct, and your prayer is valid. There is no harm in not performing the prostration of forgetfulness (sujood al-sahw), based on the hadith: "If one of you stands up from two rak'ahs, and has not yet fully stood up, he should sit down. But if he has fully stood up, then he should not sit down, and he should perform two prostrations of forgetfulness." And the hadith: "And if he has not fully stood up, he should sit down, and there is no forgetfulness (sujood al-sahw) upon him." The matter is broad, so whoever prostrates or does not prostrate, there is no harm.