Does Photo Stream update later if computer is offline?
1) In my experience, I've found that simply quitting either Aperture or iPhoto and re-opening is the easiest solution to a stall in the syncing process. I've experienced similar issues before. As to to the "why?" - that being "why does Photostream stall when switching networks," I can't provide an answer. But the failsafe solution is to quit and re-open the application.

2) Again (in my experience - Apple's documentation on this is sparse), I've found that photo stream only actively syncs when the machine is 1) awake or 2) sleeping, but not 'hibernating.' Unlike iOS devices, it seems as though Photostream is more stubborn in regards to OSX devices regarding the active and continuous syncing of data. 

3) Yes. Either iPhoto or Aperture need to be open.