It might be easier to explain if we skip over the various guitar styles and go to the extreme of something like a violin, or upright bass. In these types of acoustic instruments, the object is to transfer as much energy as possible through the bridge and into the body, because that's the only thing that makes significant audible sound. This is exactly the opposite of a solid body electric, where the only sound that counts is what the pickups can get from the string.
With a violin/cello/etc. the bridge has two feet, and you put a soundpost under the treble-side foot (braced between the front and back body plates) to act as a fulcrum, or pivot-point. As you then bow the string (sideways), the bridge is designed to rock with the vibrations, through an increasingly greater arc of travel as you move from the treble to bass strings (there is no brace under the bass side foot).
The bridge is deliberately quite tall on these instruments, because the downforce you get from the strings assists with the transfer of vibrations from the string to the top plate. The bridge itself must be carefully designed - and even tuned, by shaving and such - to transmit an appropriate blend of frequencies to the body.
(In fact, research on good sounding violins indicates that their bridges actually emphasize or enhance frequencies in a gentle peak centered around 3000 Hz, which helps to counter the fact that energy loss in the wood top plate increases rapidly above around 3500 Hz, due to frictional damping both across and along the grain... but I digress.)
An important point to note with this family of instruments is that you don't really care about sustain much at all - as long as you keep the bow moving, you generate more energy, and everything is designed to get as much of that out of the string and into the body. You can make the note last as long as you want, but it really isn't the same thing as the sustain we talk about with electrics.
Now if you want a plucked note to sustain on a solid body, you take the exact opposite approach and try to keep all the energy in the string. The nut and bridge (and whatever supports them) should appear pretty much like a brick wall, so that when the wave travels down the string and hits the bridge, it has no choice but to bounce off it and be reflected back up the string. A massive bridge block helps tremendously in this regard.
I don't know enough to say much about all the variations of guitar construction, except that they're different... On a fully acoustic guitar, you definitely care about the different plate vibration modes (here the wood, grain, and orientation are particularly relevant), though I'm convinced that the big round sound hole has a different purpose or behavior than the F-holes on a violin. The F-holes are carefully sized in relation to the volume of the air cavity, and positioned relative to certain plate vibration modes, in a way which mostly affects a very limited range of frequencies. My guess is that the sound hole on a guitar behaves a bit more as an echo or amplification chamber, but I'm really just guessing at that. I have no idea why acoustic guitar bridges are usually mounted flat on the top plate, other than the fact that it's a convenient and simple construction technique.
Regarding PS1, my understanding is that Badass bridges came into being mostly because the old Fender bridges (for which they just happen to match the screw holes...) were basically flimsy pieces of junk. The Badass is much heavier, the saddles can't move from side to side, etc. so you get a more rigid bridge and better sustain. (Others may be able to offer more insight here, I've looked at them but never tried one.)
As for PS2, I haven't seen these, but I expect it's a simple concept of mass loading. Way out there at the end of a long neck, the peghead is going to have a tendency to vibrate; if you load it up with brass blocks or whatever it will be harder to set in motion (at least for most of the frequencies you care about).
Going back to the violins again (I hope this will be interesting to some), it turns out that there are two very strong resonant peaks which play a critical role in the overall timbre of the instrument: a main air resonance peak which comes about from the body cavity and f-holes, and a main wood resonance from the overall body itself. The really interesting point is that these resonances line up very closely with the frequencies of the middle two open strings (air for the lower, wood for the higher). There's a good diagram of this here:
http://hyperphysics.phy-astr.gsu.edu/hbase/music/viores.html I happen to be personally interested in getting as close to the sound of an upright as practical, and one of the reasons I got an SF-2 was to try to approximate some of this behavior for an electric bass. For my purposes, I also believe it is possible to have too much sustain (gasp!), which is why I intend to experiment with bridge blocks of various types of wood - not just to reduce sustain, but to do so (somewhat) selectively for various frequency ranges.
Enough for now, I should go do some work.
