Why did he tell Haman to hurry?
Vilna Gaon: Soon will be the Mishteh. The king wanted to be able to tell Esther that he already rewarded Mordechai.
Malbim: He was angry that Haman was so haughty to want to use what the king used. Adoniyahu asked for Avishag, and Shlomo said that this is Mored b'Malchus, to request what the king used, and killed him for this (Melachim I, 2:22-24)! Haman should rush to do so to Mordechai, to whom the honor belongs.
Why did he tell Haman to do so himself?
Vilna Gaon: He thought that also Haman wanted to honor Mordechai. Haman himself will do so for Mordechai, and I will be exempt [of giving the reward due to him].
Malbim: The grandeur that belongs to Mordechai, Haman took it via guile. He should run in front of Mordechai like a slave!
Why did he say need to say "Mordechai ha'Yehudi ha'Yoshev b'Sha'ar ha'Melech"?
Torah Temimah citing Megilah 16a: Achashverosh said, do so to Mordechai. Haman asked, which Mordechai? He answered, "ha'Yehudi"! Haman said, many Jews are named Mordechai! Achashverosh added "who sits in the king's gate."
Why did he add "don't omit anything that you said"?
Torah Temimah citing Megilah 16a: Haman said, it is enough to give to him (rights to collect taxes from) a village or river! Achashverosh said, give to him that also - "don't omit
