Does the window for the Ubuntu One client need to be actually open (and in my Windows or Ubuntu "taskbar") for the sync'ing to actually occur?
It would be nice if the sync process could occur without having the window always lying around. Dropbox doesn't require a window to be open, for example.
No, it does not require the client window to be open. Ubuntu One has a sync daemon (a process that runs in the background), and you can close the client window and it will still sync files automatically.
The only place where this is not true is on iOS, which limits the background activity of apps.
On Windows there was a bug before 2.0.3 that meant that syncdaemon didn't start automatically, but 2.0.3 was released on December 19 so everybody should have it by now.
Not on Ubuntu, but yes on windows. Technically, you only need to have ubuntuone-syncdaemon running, but the "normal" way to start it on Windows is by getting the icon on the taskbar's notification area.