Who are "Chachamim Yod'ei ha'Itim"?
Torah Temimah citing Megilah 12b: "Chachamim" are Rabanan 1 . "Who know the times" - they know how to make leap years and to fix Rosh Chodesh.
Vilna Gaon: There are two courts. One judges based on letter of the law. The other judges Hora'as Sha'ah - at times one must be lenient, and not judge based on letter of the law, e.g. for a king or great person. They are lenient, amidst concern for his honor. He wanted to save her, due to her lineage, and there was no one in the world as beautiful as her. However, he could not save her, for he feared lest the nation rebel against him!
Malbim: Matters of the king require those who know how to judge according to the time. One who sins against the king when he sits on his throne is unlike one who sins against him when he passes from place to place. All the more so it was severe to sin against him when all greats of the Medinah were there! Anyone who knows laws can judge between colleagues. The king told Yod'ei ha'Itim that this case is like the latter. She is a queen from birth; she sinned against her colleague, her spouse. In spite of his anger, he wanted to save her; love covers up for all sins. Also refer to 1:13:3:2.
12b: Achashverosh asked Rabanan to judge her. They reasoned - if we tell him to kill her, tomorrow he will sober up and blame us for convicting her. We cannot tell him to pardon her, for she disgraced the king! They said, from when we were exiled from our land, we lost the counsel needed for capital cases.
What is "Devar ha'Melech"?
Rashi: Every Mishpat is brought in front of people who know laws.
Vilna Gaon: The king himself does not judge Devar ha'Melech - something that pertains to himself.
What are "Das va'Din"?
Vilna Gaon: Das is knowing the law. Din is knowing how to judge according to the claims.
Malbim: Das is usually religious laws, but it can refer to ethics. Even a wife who disobeys her husband should be punished. She must do his will. However, women of Paras and Madai were modest 1 , and stayed in their homes. They did not appear in front of men - 'Araviyos go out with their faces covered; Madayos go out with a cloak buttoned [closed]' (Shabbos 6:6). Therefore, Vashti is not liable 2 .
Vashti was Bavlis! Perhaps Malbim explains why the judges, who were of Paras and Madai, should exempt her, for they are used to modest women. (PF)
Malbim (22): Why does the Megilah tell about the extent of his rule and wealth [and his desire to exempt Vashti]? It shows the miracle that occurred. It is feasible for a beautiful woman to defy a king - but not if (a) he ruled over 127 Medinos, (b) conquered them so quickly, (c) was so wealthy, (d) was so wise to get the people to grant him unlimited kingship, and (e) he loved her so much. Even so, his love of good (that women not disgrace their husband) or benefit (unlimited kingship) overcame his love of pleasant (Vashti). Surely his love of pleasant for Esther would not make him recall the letters - rather, it was from Hashem.