Why did Avimelech tell Sarah that he gave Avraham a thousand silver pieces, when in Pasuk 14, the Torah relates that he gave Avraham a list of gifts without mentioning money?
Rashbam and Seforno: The money mentioned here refers to the money that he gave Avraham as betrothal money. 1
Rashi and Ramban: The thousand silver pieces is the value of the gifts that he gave Avraham. 2
What was Avimelech telling Sarah? What was the covering of the eyes that he mentioned?
Rashi and Rashbam: He was appeasing her by pointing out that he had not returned her to Avraham empty-handed. Rather, out of Kavod for her, he gave him expensive gifts, thereby covering over the eyes of the members of their household from viewing her with contempt and restoring her esteem in their eyes, 1 since they would now realize that he was forced to return her due to Divine intervention.
Ramban: The gifts that Avimelech gave Avraham 2 (the fact that the king needed to atone for sinning against the wife of a prophet) would deter others (cover their eyes) from looking at her or at any member of her household. 3
Seforno: The "K'sus Eynayim" was a garment of many colors that was a symbol of Kavod. He was pointing out to Sarah that he did not take her for prostitution or as a concubine, but as a wife (to become queen, to boot).
Targum Onkelos and Targum Yonasan: The money that he gave Avraham was to compensate the night that he 'covered (hid) her' from her husband and gazed upon her, (although even if he gave her everything he owned, it would not really suffice to dispel the shame that he had caused her
The Rashbam explains the Pasuk like Rashi, only the Kavod Avimelech referred to lay in the exceptionally large amount of the betrothal-money that he gave Avraham.
The Ramban also suggests that "Elef Kesef" means many thousands of silver pieces, It is not clear however, if it still refers to the above-mentioned gifts that he gave Avraham, or to an additional gift that the Torah did not mention.
In which case, Sarah's being taken by Avimelech turned out to be a good thing for Avraham and her.
We learn from this that a commoner's curse should not be taken lightly.
What does the final phrase in the Pasuk "ve'es Kol ve'Nochchas" mean?
Rashi: She would now be able to explain to all and sundry what he had just told her, and they would believe her due to the large gift that he had given to Avraham.
Rashbam: The gift was a clear proof that what he said to Sarah was said in honor and not in rebuke.
Targum Yonasan: The money that he gave to 'her brother' would prove to him that he had not been intimate with her.
Da'as Zekenim, Hadar Zekenim, Rosh: To all who do not know that he is your husband, take rebuke - do not say that he is your brother, lest it lead to sin!
What were the "Elef Kesef" that Avimelech gave to Sarah?
Targum Onkelos: They were Sela'im 1 (a Sela is four Dinarim).
Hadar Zekenim: "Elef" is an expression of teaching. Or, it is [cattle,] like "Shegar Alafecha" (Devarim 7:13). Kesef is an expression of desire - "Nichsof Nichsafti."
Moshav Zekenim: Kidushin 11a says that Stam money mentioned in the Torah with a fixed amount is Dinarim of pure silver! That is only when the Torah obligates an amount. Here, it merely tells what Avimelech gave.