An excellent choice!
My take on Hedge Traditions is that they are sketched out, rather than fully developed. No slight on the developers here! HMRE describes a series of magical traditions, cramming rules and fluff for each into a short chapter. These traditions are not played very often. By contrast, Hermetic Magic has undergone 5 editions of refinement and enjoys thorough and ever-increasing coverage.
This means that you both get to and need to decide a lot for yourself. BTW, if you successfully pull together this saga and run it, you may soon become a better Authority on LMs than the people who designed it. Even if not, the usual GM prerogatives hold. So your answers to these questions are the best.
I like the idea of LM. I imagine diverse traditions with a lot in common. Since the liberal arts and philosophia are supposed to be universal, there may be LMs of some sort wherever there is advanced education, from Egypt to India to China, all using magic based on a common foundation. Of course, the (not canonical) LMs of Salamanca use AL(music) for their magic, and their common major virtue reflects their ability to bolster their magic with music (and maybe have Duration: Performance) rather than the ability to reduce hostile Auras; it is not for nothing that the University of Salamanca offers a degree in music!
I suspect that LMs of different traditions would have to put some effort into understanding the others' methods and texts. Perhaps some LMs can easily understand each other, but other LMs have experiences that are too different to even allow recognition that their magics are the same under the skin. All up to you.
Do LMs of some traditions form an association that is far more widespread than the OoH, though less powerful and less formally organized? Do some academic institutions have so many LMs, that there are enough to have antagonistic factions and sub-traditions?
So many things to decide.
You also get to decide how powerful you want LMs to be; the rules technically allow LMs to be very, very powerful, though I suspect RAI would have them be pretty pathetic. For example, the best way to obtain protection is usually to use Fo or Ma Forms to grant appropriate virtues (major immunity ftw) rather than the expected way. Similarly, although LM cannot itself directly do stupendous things, it can grant major and minor virtues that allow LMs to use Powers, Shapeshift, and so on. Magicam is a fantastic Form. I like LMs strong, but YMMPV.