It is a delicate balance between finding the best frequencies, having the right antennas, adjusting the squelch. Without a spectrum analyzer, I would suggest you do it the hard way - once you get the antennas and squelch settings the best you can, every time you have an issue with a mic, write...
If you buy a digital mixer, it is important that the teacher be willing to make the effort to learn how to work the board so they can teach the students. Most of he 11 high schools around me bought Yamaha LS-9 boards several years ago, one has an older Presonis (sp?), one has a Soundcraft...
The photos look like the failure of the blue wire and the other black wire were both right at the screw connection. If there was any plating on the screw or the rest of the socket it is long gone - I would replace the socket, use ferrules. 12ga wire is fine.
Your main problem is not really the strength of the wire, the strength of the connection, or how you are taping the mics, as it sounds like you are taping them correctly. While some brands of these mics are a little more robust than others, all of them are fragile. And it is not so much the...
The best stuff I have found for this is 3M General Purpose Adhesive Cleaner 08984. Leaves no residue, does not seem to hurt the cables. It is normally used by body shops to remove the adhesive that holds your car together...buy it at an automotive paint supplier...