Why did Avraham instruct Eliezer to place his hand underneath his thigh?
Rashi: Because an oath requires the person swearing to hold up his hand an object of Mitzvah (such as a Seifer-Torah or Tefilin), and the Bris Milah, which was the first Mitzvah that he had been commanded, was particular dear to him.
Rashbam: This is an act of subordination that we only find performed by an Eved to a master or a son to a father 1 .
Ha'Emek Davar: This was to make a Bris with him, in addition to the Shevu'ah.
Why did Avraham instruct Eliezer to put his hand under Avraham's thigh, and not under his own?
Oznayim la'Torah: Because Avraham's own Mitzvah was dearer to him than Eliezer's.
What is "ha'Moshel b'Chol Asher Lo"?
Yoma, 28b: 'He ruled over his master's Torah
One may not touch the Ever, lest it causes emissions (Nidah 13b)!
Moshav Zekenim #1: This applies only to youths, but not to old men.
Moshav Zekenim #2: It is forbidden only past the crown. Eliezer held it before the crown.
Ibn Ezra, Radak, Chizkuni: He put his hand under the thigh, not on the Ever.