I understand that notifying the user that there are new answers for the current question is really important, but why doesn't the site provide "live" information also for new messages in inbox (comments, answers etc..) and for reputation change?
The notification bar would become too cluttered, and frequently ignored, if it popped up for every reputation change. The little envelope icon next to your username at the top of the page lights up for such changes, and useful changes on all sites highlights the "StackExchange" drop down link in the upper left, so those alerts are alreay provided, just not as obviously as notification bars.
I would guess it is lower in the order of importance and possibly more expensive to keep a live update of.
Getting answers to your questions is the primary use case for Stack Overflow, so getting notified of new answers is given the highest precedence. Comments and reputation are nice features of the site, but they're really only secondary to getting answers.
If you are interested in getting notification about your rep change have a look to Stack Apps notification tag; there are plenty of fancy app[s] developed for this specific purpose.
The cool thing of these apps[s] is that you can get rep change notification even without browsing Stack Overflow.