Why did Eliezer go to the well to test her, and not to the house of Avraham's family?
Riva: In the house one can investigate only beauty and lineage, but not Zerizus and generosity - perhaps she acts due to her father's command.
Ohr ha'Chayim (12): If he would go to Avraham's family, they might give to him a Shifchah, and say that she is from their seed. Rather, he went to the well to test that she does more than she is asked, and asked about her family before she knew the reason.
What were Eliezer's criteria for picking a girl?
Malbim: He would approach the nicest looking girl, and test her Midos. To make the test harder, (a) he stood by the well, so she could say 'you can take by yourself!' (b) He would ask after her jug was on her shoulder - so she could say "ask someone whose jug is still in her hand!' (c) He would ask her to tilt her jug - so she could say "tilt it yourself
We do not find that he told the men with him to go away. Perhaps she would assume that they are tired or weak from traveling in the sun, or are lazy slaves. (PF)