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I remember learning some years ago in Gemarah Megillah 13: that when Haman drew lots he was happy it landed in Adar since it is the month that Moshe Rabeinu died. However, he didn't know that Moshe Rabeinu was born in that month as well.

1. How and why would Haman know the birthdate of a Jewish leader? (i.e. why would a gentile care to know Jewish trivia?)




2. Say Haman knew it from some Jewish sources,
   Wouldn't it make sense for someone who heard, that on the 7th day of Adar is Moshe Rabeinu's Yahrzeit, would hear in the same sentence, that on THE VERY SAME DAY was Moshe Rabeinu's birthday?

The מהרש"א on that Gemara explains that Moshe having died on the 7th of Adar is something which could be worked out from Scriptures, as the Gemara works it out on Kiddushin 38a (it says they wept for Moshe for 30 days, then Yehoshua told them they had three days before crossing the Yarden, and they came through the Yarden on the 10th day of the 1st month, so just work backwards and you get the 7th of Adar).

However, Moshe being born on that day is only learned through a Rabbinical derasha, and that was not known to Haman.

In terms of your first point, it is not that unusual to find in Rabbinic tradition the enemies of the Jewish people being strangely familiar with Scripture.