Why wasn't the first party co-ed?
I don't know enough about Persian culture at the time. This might be a problem. To cite Herodotus (father of history or father of lies):
"Dear Macedonian, we Persians have a custom when we make a great feast to bring with us to the board our wives and concubines, and make them sit beside us. Now then, as thou hast received us so kindly, and feasted us so handsomely, and givest moreover earth and water to King Darius, do also after our custom in this matter."
Then Amyntas answered - "O, Persians! we have no such custom as this; but with us men and women are kept apart. Nevertheless, since you, who are our lords, wish it, this also shall be granted to you."
But what is the idea here in Megillah? Perhaps Achashverosh's party was a diplomatic one while Vashti's served a purpose of keeping her close female friends entertained. Or maybe the riotous drinking did not make it a conducive environment for the women. It might, or might not be a very good question.
As someone pointed out in the comments, the Greeks may have been involved in their own wars. Besides any teretz, who is to say that there were not generals commanding their troops out there.
Why didn't the king blink when Haman asked for permission to murder millions?
He wasn't such a great guy. And he did not put an exact figure on it. It was a possibly despised minority, and Haman took pains to show they were not valuable, nor were they patriotic. This is in the text of the megillah itself. You might regard this with doubt, but it is not, IMHO, a "plot hole."
Why didn't the King take Haman's bribe? After throwing all those parties, he didn't need some spare change?
No. The purpose of the parties was to exhibit his great wealth. And he still had great wealth to spare. He did not need the "spare change." Haman originally thought to give this gift as a form of persuasion, but Achashverosh turned it down and showed favor to one of his top advisors. What is the question, even?
Why did Haman need to build a gallows for Mordecai? Couldn't he just hire the firm of Two Guys Named Vito to settle the score quietly?
The megillah itself states that he did not see fit to just take on Mordechai, but intended to take down all of Mordechai's people as well. And so he went to the king and arranged for all Jews to be wiped out. Quite possibly, he intended to use the gallows on that day appointed for killing all the Jews.
To hite the firm of "Two Guys Named Vito?" Mordechai was a member of the royal household. He lived in Shushan HaBirah, the citadel of Shushan rather than the city. And to hire the equivalent of Bigtan and Teresh would be underhanded. If they were caught, they would finger Haman, and he would get in trouble. Why should he bother to do this when there is an entirely above-board solution to the problem, via official channels, already in progress. Again, I don't see this as a "plot hole," or even a question.
Why didn't Esther accept the King's offer of half the kingdom. He made it twice, so we can assume he was serious.
Or else it was just an expression, as we see it repeatedly in use. Half the kingdom, anyway, would not have been a solution to the murder of Jews throughout the entire kingdom. In the end, as the favored queen, she effectively ruled over all the kingdom.
Why did Esther drag Haman and the King to a second party? Couldn't she have ratted out Haman at the first party?
People are people, not robots. She might have been reluctant at first, or frightened he would not grant it, favoring her over his favored minister. Don't read nuance and personality as a plot hole.
Why did Charvonah have such a big mouth? And if he had it in for Haman, why did he wait until the last minute?
This is classic politics, not a plot hole. If he had it in for Haman, he might not have had anything to say beforehand. He waited until they covered Haman's face to put in his two cents. And who knows, that might even get him favor from Achashverosh and the newly powerful Esther (and perhaps Mordechai).
Why did Mordecai get Haman's job? What were his qualifications? Is that how it worked in Persia? If someone plotted to kill you and your people, did you automatically take his position when the plot was foiled?
No. But he already lived in the palace (in Shushan HaBirah), and was related to the queen, Esther. He was set over Haman's house. And being close to the king, he attained some power, such that he became מִשְׁנֶה לַמֶּלֶךְ.