Lets see if I can rough these in.
Though on the whole Sun-pools seem to function randomly, is there a larger pattern at work such as the location of other celestial bodies within the solar system that people haven't seen simply because they haven't been looking / don't have a big enough data sample to recognize it?
The paths connecting pools are believed to be the direct intervention/choosing of the Sun-god Avva. Thus sunchasers are treated a being guided by divine providence and their arrival has a hint of destiny to it. Avva may
make these decision based on other celestial phenomena, but as you say, no causal link is currently known to mortals.
What exactly are moon-pools and how are they related to sun-pools?
Unlike most cosmologies that are Sun-centered, Sunchaser has a Moon-centered larger universe. There are 5 worlds plus the Underheart in the larger system and while each world has its own Sun, it is the same moon that shines down on all of them. Moon-pools act largely like Sun-pools but connect different worlds (and at one time the Underheart) in the same way as the Sun-pools connect places under a single sun. They are considerably rarer, and open much less frequently.
What does it take to break or destroy a sun- or moon-pool, and are they self-repairing? What would it take to create one? What effect would either action have on the larger system?
Filling one in or dredging one (too deep and they don't open either) could disrupt one. They are self repairing only in that the forces that generate a natural pool are difficult to restrain over long periods of time. Ponds an other bodies without an active outflow generally do not serve as sunpools, so you need a lot of running water to qualify.
How does the travel between them work once you're "lost in the glare"? Is space folded pool-to-pool, or is there some sort of hyperspace-like movement involved once the link as been established between pools?
Generally you just walk forward until you come out of the glare (which some believers claim is the direct gaze of Avva upon you - though he's not chatty). Trying to circumvent the pools as two way-travel with things like tying together with ropes generally results in you not entering the glare at all (unless the entire linked packet is small enough to fit into the gleaming region in the midst of the pool). The total distance walked is usually slightly larger than either pool could encompass, so a "hyperspace-like" middle region is probably the more accurate model.
Do the pools of either sort exist outside the Valley?
Yes. Both versions are available as explanations for why the GM might introduce characters outside of the normal spread for the setting. Sunpools could draw in resident of other parts of the world while moonpools could provide for cross-campaign visitors.
I anticipate the existence of spells (possibly prayers considering the mechanism at work
) That can set the destination of a pool about to open (6th level?), divine where a pool is going to lead to (4th level?), and open a random pool outside of the normal window (3rd level?). Snowblade arguably enacted a spell that locked a moonpool at the cost of her own life, so lesser version of that effect might also be possible.
The sun-pool "network" is a capricious thing - its not been pressed into service as a trade corridor because of its extreme unreliability in destination. It also has a strong reputation for "there's no going back". Its very rare outside of the use of powerful spell/prayers that a sun-pool jaunt will ever take you back to a place you've already emerged from once even in a lifetime of travel. It may be possible that people essentially carry an internal register of all their previous jaunts and that register is used as an exclusion list when a pool cycles open...