I think I understand the general idea behind controlled directivity, but I don't get:
Very good list of questions:
- Why a compression tweeter in a CD mount would sound better than a more typical tweeter on a flat baffle.
I don't know that it's really the compression tweeter that makes it sound better, it's the waveguide. Waveguides can be effectively used with dome tweeters, as well. What they allow is very wide and constant dispersion. Typically, a dome tweeter will do well on axis and even do pretty decently off axis to some degree, but their off axis performance will always differ from low frequencies to high. They will have very wide dispersion at lower frequencies that diminishes as the frequency goes up. The problem with this is that the reflected material in the room will have a different sonic character than the direct sound. I don't know if there is individual sensitivity to this, but I've often heard this contribute to the head in the vice quality that many speakers have (including many horns).
- Why PA speakers better suited for large auditoriums would sound better that home audio speakers in a home setting.
I'm not sure they are better suited for large auditoriums compared to small rooms, as opposed to better suited to auditoriums than smaller speakers. I think there is an important distinction between the two. While a small 2 way is unsuited for an auditorium, a 15" woofer isn't necessarily unsuited to a small room, unless of course, you are trying to cram some 30 cu ft speakers into a 500 cu ft room.

PA speakers can have advantages in terms of power compression, as well, for those listening at a high level. I know I go a bit over the top, but when I design speakers, I tend to make what I refer to as 20 or even 30dB speakers. I want to build a design that is capable of peak output significantly higher than my average level. (I go over the top and typically try for 30dB). This doesn't mean I listen loudly, but I do have a couple of pieces of music with 25-28dB crest factor and if I want to turn the volume level up, I don't want my speakers collapsing.
- Why anyone would try and use a 15" woofer in a 2-way even if the tweeter might go as low 1 or 1.2 kHz.
As I mentioned in my last post, I'm not sure this is an ideal matchup, but because of the specific driver chosen, I bet it fairs very well. If someone wanted to use an Eminence or JBL in that same application, I may caution against it. For whatever reason, the TD-15M does a lot of things very right.
I think the last point to make about pro audio drivers is that they often exhibit measurably better distortion performance than home audio equivalents.
I also believe that a healthy dose of skepticism is important and would encourage anyone who gets a chance to hear for themselves what a high end modern horn system (using quality waveguides) is capable of. It still may not be your thing, but I suspect you may be surprised.