What is the meaning of "Hegi'a Devar ha'Melech v'Daso Lehe'asos"?
Malbim: Until now, the exposed part of Haman's letters said only to be ready for Adar 13; that of Mordechai's letters said that the Jews will have the upper hand. On Adar 13, both letters were opened. Also, this was the day of action. The two sides could not compromise and do nothing; the king's edict mandated fighting!
What is the difference between "Oyev" and "Sonei"?
Vilna Gaon: An Oyev wants to do evil himself. A Sonei rejoices to see the evil [of the one that he hates], but he himself does not do anything. Therefore, it says that the Oyevim hoped to rule over them. It was reversed - the Jews ruled even over Son'eihem.
Why does it say "Yishletu ha'Yehudim Hemah b'Son'eihem"?
Vilna Gaon: The 49 gates 1 were for evil for Yisrael, and all were reversed to good, to rule over their enemies.
Malbim: On this day, Yisrael were at the lowest Mazal, and Amalek at the highest. This is why Haman chose this day! The day itself caused them to hope to rule over Yisrael. Hashem's Hashgachah reversed Yisrael to the highest Mazal, and Amalek to the lowest. This was not natural. Do not say that others with good Mazal helped Yisrael - "Yishletu ha'Yehudim Hemah b'Son'eihem", by themselves!
Magihah in Perush ha'Gra ha'Shalem (from his Perush on Mishlei): There are 50 Sha'arim of understanding; 49 were given to Moshe. There are corresponding gates of Tum'ah. Ohr Chadash (6:11) - Haman wanted to reach the 50th. Yisrael overcame him on the day of bringing the Omer, when we begin counting 50 days until Matan Torah.