1) I stated in the podcast that the walls of Jerusalem were beached on the 17th of Tammuz.  All agree that this is commemorated on this day, but the actual date of the breach is somewhat unclear. 

Jeremiah (39:2) seems to set the date on the 9th of Tammuz, while the Mishnah in Taanis says the walls were breached on the 17th of Tammuz. 

Both the Jerusalem Talmud and the Babylonian Talmud ask this contradiction and resolve it differently.  The Jerusalem Talmud says that the verse in Jeremiah made the wrong calculation and it really DID happen on the 17th of Tammuz, like the Mishnah says.  However, The Babylonian Talmud says that the verse in Jeremiah is referring to the first temple (the one discussed in this episode), while the Mishnah is referring to the walls being breached during the time of the second temple. 

So in the end, there’s a debate between the Talmuds as to when historically the walls of Jerusalem were breached in the times of the first temple.  Either way, all agree that we commemorate it on the 17th of Tammuz, as that’s when it was breached in the times of the second temple. 

For more analysis on this see the explanation of the Maharsha (Hebrew only).

2) Our calendar is se in a way that it is impossible for Rosh Chodesh Teves to fall out on Thursday, and it’s therefore impossible for the 10th of Teves to fall on Shabbos.  The opinion of the Abudraham and his opponents therefore isn’t practically relevant. However, the idea that it would teaches us a great lesson about the nature of the 10th of Teves. 

Additionally, none of the other fasts can fall out on Friday; the 10th of Teves is the only one that can practically fall out on Friday.

3) There are two other reasons, in addition to the siege, that are mentioned in Jewish law for the fast.  On the 8th of Teves, King Ptolemy demanded that 72 Rabbis translate the Torah into Greek.  This is famously known as the Septuagint, and the miracles that occurred with it are documented in the Talmud, Megillah 9a.  Our sages viewed this translation as a tragedy for the sanctity of the Torah.  Indeed, the Septuagint is canonized in many Christian branches. 

Additionally, on the 9th of Tammuz, Ezra the Scribe passed away.  Ezra was one of the greatest Jews in history, who restored Judaism to a lowly nation at the time of the beginning of the second temple.  The Talmud in Sanhedrin lauds Ezra as befitting to bring the Torah down to the Jews, had Moses not already done so.  Although these two tragedies happened earlier, the sages included them in the fast of the 10th of Teves.