Why does it say "after these things?"?
Rashi citing Megilah 13b: [Haman was raised] after Hashem created the cure 1 (a reason for Achashverosh to love Mordechai, which led to Yisrael's salvation) for the malady. Hashem does not strike Yisrael unless he creates the cure first - "When I cure Yisrael, (afterwards) the sin of Efrayim is revealed (through the punishment)" (Hoshe'a 7:1).
Vilna Gaon: Memuchan is Haman (Megilah 12b). His counsel killed Vashti (which led to Esther becoming queen), and enabled the king to judge by himself what pertains to him. If not, it would have been very hard to judge Sarim greater than the five Ro'ei Pnei ha'Melech 2 .
Malbim: After Achashverosh forgot that Esther told him in the name of Mordechai, he wanted to bestow good to those who benefited him. He did not find a way to pay Esther for saving him from death. He thought it would be nice for her to benefit the Sarim who counseled him to take Esther. The foremost was Haman, who counseled him to seek Besulos. Haman was succeeding; he was able to convince the king that he told him, therefore the king raised him with the honor due to Mordechai. Hashem decreed so, until the crown would return to its owner (Mordechai deserved the honor).
Vilna Gaon The Gemara learned from "Ki Chen Tzivah Lo ha'Melech" (2). A Midrash says that "Melech" Stam (without Achashverosh) refers to Hashem! Therefore, it explained 'after Hashem created the cure.'
Melachim II, 25:19 mentions five. Above (1:14), it mentions seven! Rashi (Megilah 23a) - five of them are important.
Why does it say both "Gidal" and "va'Ynas'ehu"?
Vilna Gaon: Gidal is with money - "va'Yigdal ha'Ish va'Yelech Haloch v'Gadel 1 " (Bereishis 26:13). Va'Ynas'ehu - he aggrandized him.
Malbim: Royal etiquette does not allow suddenly raising a commoner from a low level to the highest level, rather, level after level. Also Pharaoh raised Yosef level after level, like I explained in Bereishis (refer to Bereishis 41:40:151:1, 41:41:1:2). And so Yosef said - "va'Ysimeni l'Av l'Pharaoh", then "ul'Adon l'Chol Beiso", and after "u'Moshel b'Chol Eretz Mitzrayim" (45:8). So Achashverosh first made Haman a Sar, and afterwards raised him level after level.
Why does it say "ha'Agagi"?
Vilna Gaon: This teaches that the cause was [Sha'ul] leaving alive Agag.
How was he above all Sarim?
Vilna Gaon: All kings do not sit among the Sarim, rather, in a place above them. So he made a place for Haman above the Sarim.
Malbim: He was head over all the Sarim.
