It's probably late for your gig, but I'll answer anyway --
1)
What exactly happens inside that adapter cable?
To answer that, you first need to know what a balanced line is and how it works - I won't go into details, you can find plenty of information on the web about the subject. A nice article about balanced lines, grounding and so on can be found on the our website:
Audio arcane mysteries
The balanced line was created to offer better protection from outside interference for the carried signal. This is accomplished by transmitting two versions of the signal: the normal signal and a polarity-inverted signal. When an EM interference is induced in such a line, both signals are influenced, and when the electronics at the other end sum the signals, the interference will be mostly canceled. A more complete explanation of how this happens can be found in an article made by Ian Gregory which can be found on
his website.
Returning to your question, an adapter cable from XLR (balanced) to TS (unbalanced) will take only the hot signal and the ground (pins 2 and 1 according to
EIA Standard RS-297-A), leaving the cold signal (pin 3) unconnected (open). You wouldn't want to shunt it to the ground because that would short-circuit the output differential amplifier on the mixer - not a problem mostly, since these op-amps have protection against it, but still not a wise idea for an adapter.
2)
Is this my correct connection between these two devices (XLR > TS)?
As far as I can tell, yes, it is. Just to be sure: you have two output channels through balanced XLR (L&R), you use an adapter for each channel (XLR to TS) and then combine them to form a TRS output mother jack to which you connect the BOSE stage system. Best would have been if you had stage equipment that has inputs for balanced audio, that will reduce the EM interference from your snake.
3)
What happens if I use XLR > TRS into that 1/4" unbalanced Input?
The TRS connector can be used to carry wither stereo or balanced signal. If you use XLR to TRS adapter for one channel (L or R), it will carry the balanced output for the respective channel and that is not a stereo signal. The 1/4" unbalanced input is a stereo input - it's two unbalanced channels. If you connect those together, nothing essentially bad will happen, but you'll have only one channel through both speakers, with the signals inverted in polarity.
I hope this will clarify your questions for future gigs and will make good use of our equipment
Cheers!
Sources, further reading:
Huurdeman, A. A., "The worldwide history of telecommunications", Wiley-IEEE, 2003.
Ian Gregory, "Balanced and Unbalanced Audio", University of Manchester, 2004
Graham Blyth. "Audio Balancing Issues", Soundcraft, 2009
B. Whitlock, "Balanced Lines in Audio - Fact, Fiction, and Transformers", Journal of the AES,
Vol 43, No 6, June, 1995.
Bill Whitlock, "Interconnection of Balanced and Unbalanced Equipment", Jensen Application
Note AN003
B. Whitlock, "Answers to Common Questions about Audio Transformers", Jensen Application
Note AN002
IEC, "Sound system equipment" (Third edition ed.). Geneva: International Electrotechnical
Commission. 2000. pp. 111. IEC 602689-3:2001