Does a worshiper's leaning on his hand or the like when rising from the Tashahhud for the third Rak'ah, or from prostration for another Rak'ah, affect the validity of his prayer?

Prayer is not invalidated by resting on the hands when rising from prostration. There is a difference of opinion among scholars regarding the preferred method: the Hanbalis believe that resting on the knees is preferable, except out of necessity such as illness or old age, while the Shafi'is and Malikis believe that resting on the hands is better. Ibn Qudamah explained that rising should be done on the balls of the feet, resting on the knees without the hands, citing the hadith of Wa'il ibn Hujr, Ibn Umar, and Ali (may Allah be pleased with them). The hadith of Malik ibn al-Huwairith, which indicates resting on the hands, is interpreted as being for situations of difficulty. If rising without resting on the hands is difficult, then there is no harm in resting on them.