Why did she say "if I found grace in your eyes" and "if it is good for the king"?
Vilna Gaon: One who requests needs two 1 conditions. (a) The giver wants to give it. (b) The requester has grace in the eyes of the giver, that he wants to give to him.
Below (8:5), he requires a third, that the matter is proper to do! (PF)
What are the requests "Nafshi bi'She'elasi" and "Ami b'Vakashasi"?
Rashi: I should not be killed on Adar 13, and my nation should not be killed. Vilna Gaon - "Nafshi" is She'elah (for herself), and "Ami" is Bakashah (for others, i.e. her nation). Malbim - she does not request wealth and honor, only to be saved from death.
Malbim: According to Meforshim who say that She'elah is the request itself and Bakashah is the purpose, if one asks to save another's life, the She'elah is saving the life. The purpose is because he cannot bear to see the evil to his friend. If so, the Bakashah is for himself. Achashverosh could err to think that her request for her nation is for herself - "Eichachah Uchal v'Ra'isi ba'Ra'ah Asher Yimtza Es Ami" (8:6). She refuted this. Saving her nation is so dear in her eyes that she would agree to die alone to save them. The She'elah is for herself, but the primary Bakashah and her purpose is her nation.