How does a single phase shared neutral circuit actually work, physically?
To your first question: the pole transformer has a split secondary.  It sorta acts like two 120V output transformers connected end-to-end, in that it enforces 120V between neutral and each hot.
So if, say, I went and turned on all the circuits in my house on one leg, and left all the others off, the voltage on each leg would be pretty close to 120V*.  (And in that situation, the neutral current would probably be as high as it could ever get without faults in the various 240V appliances).
To your second question: because the two legs are out of phase, if they're equally loaded then any current flowing into the nut from one leg is matched by an equal current flowing out of the nut to the other.  If the loads were perfectly matched, no neutral current would flow at all.
To your unnumbered third question: I assume you're talking about a panel that has three 120V legs and 208V leg-leg.  In that case, three balanced loads, one on each leg, will sum up to zero current.  Any imbalance will result in current, but since the legs are 120 degrees out of phase with each other, it's kinda complicated to figure out what the neutral current may be for any given combination (it can be done -- just make friends with Mr. Trigonometry and get to work).
* Actually, the loaded leg would have slightly lower voltage -- just close your eyes to that right now...