If a spiritually active human (say, Uryuu) died in combat, would his soul just pop out and keep fighting? (and when they were exterminating the Quincy did shinigami have to kill each one twice?)
Probably not. Given that the fight occurs on a spiritual level it is basically assumed that whatever harms the body harms the spirit too, so if you die bodily, you die spiritually as well.
What happens to the people eaten by hollows when they are purified? Do they split or do they remain part of the ex-Hollows?
Souls eaten by Hollows are Lost Forever(tm). If that weren't the case, Soul Society probably wouldn't care to hunt the hollows beyond sport.
How come most shinigami don't actually do anything?
It is heavily inferred in the series that becoming a shinigami requires extensive training and only the moderately competent are actually sent to patrol the human world. Since becoming a Shinigami in Soul Society is not unlike having a high-payment military job, you get tons of applicants... the excess has to go somewhere. It is also heavily inferred that given some Hollow are particularly hard to defeat, sending a ton of Shinigami all at once is akin to giving them a (permanent) death sentence. Ergo, they only have one, maybe two Shinigami patrolling any given area at a time.
Also (this is my conjecture, though), given that Shinigami tend to give a certain vibe and that spiritually strong people are relatively plentiful, it is implied that sending a ton of Shinigami around at any given time would increase the general spiritual strength of the populace - thus creating more appetizing meals for Hollows and driving them out of Hueco Mundo. (This is supported by the fact that Ichigo's noseing around as a Shinigami contributed to awaken Orihime and Chad's powers, as well as allow Tatsuki to see spirits.)
Do people in Soul Scoeity remember their past lives or not?
It is somewhat demonstrated in the series that new souls can actually be born inside Soul Society. More to the point, it seems to be pretty hit-and-miss. (Remember the kid living inside the parakeet that Chad saved? That particular kid remembered Chad.) It would appear that either A) people in Soul Society tend to remember, but don't particularly care, or B) some remember, some don't. Those actually born inside Soul Society have no previous life to remember, obviously.
Why/how do gigai and artificial souls work in Soul Society?
Remember when they first invaded Soul Society? Urahara had to send them bodily through a passage that materialized them within Soul Society. That said there seems to be no reason to use gigai within it, or even a way to do so (since in the real world spirits have no mass, but in the spiritual world they do). The more likely answer is simply "you don't use gigai inside Soul Society".
Artificial souls seem to be the exception to the rule. They ARE, pretty much, compressed souls in the form of a pellet made to keep the gigai safe while Shinigami are busybodying around. The general understanding seems to be that once inside the gigai, the artificial souls get to move them around by extending their influence throughout the gigai. With that said, we go back to the point that there is no use in having a gigai lumbering around inside Soul Society (as, apparently, anyone within it can tell what's a soul from what's a gigai. I suspect that's part of the reason Aizen has a plot-device Zanpakutou as well).
Artificial souls just function oddly. Outside a body/gigai, they seem to have physical substance (demonstrated by the fact one can actually pick them up bodily). Once inside, however, they seem to act as spiritual particles (demonstrated by the fact Urahara and Rukia can stab/punch the pellets out of the bodies by means of their equipment). Whether this property remains the same inside Soul Society or not is anyone's guess.
Honestly, I think the biggest question is... if Urahara is on Earth against Soul Society will (having escaped there, obviously), why is A) his presence within Karakura seem to be ignored by Soul Society at large (considering Rukia knows him by name) and B) who does he get his merchandise from, since Yoruichi seems to be in the same boat as him? The answer to B) could be "he just makes his own stuff", but that's unlikely due to the whole Kon episode, and also because he apparently gets Soul Society-brand products (as demonstrated by Rukia's insatisfaction with him not carrying Chappy-brand artificial souls).
