Seeing as "Im" always means 'if'', what exactly was Avimelech saying?
Ramban: Bearing in mind that every oath is connected to a curse, what he meant was that Avraham should swear to him that he accepts upon himself Divine retribution
Ramban: Following the Torah's standard format
What is the significance of Avimelech's reference to his 'son and his grandson'?
Rashi: He asked that Avraham's kindness should last as long as his son and his grandson 1 lived, because that is as far as a person's mercies tend to extend.
By which G-d did Avimelech ask Avraham to swear?
Targum Onkelos, Targum Yonasan and Minchas Shai: 'Elokim', which is Kodesh, refers to Hashem.
Rosh: Avimelech asked him to swear in the name of idolatry. Avraham however, responded by swearing in the name of the One who will later say "Anochi Hashem Elokecha."