Carolyn,
I have found that the best way to get around is to totally ignore the normal rules of the road. This goes for pedestrians as well as drivers. If walking - do just that, I know a few locals who are horrified by the way I step out (not too far) in front of traffic, but the simple fact is that if they have an accident, they have to wait (maybe hours!!) for the police to turn up - you as a pedestrian don't, so generally they stop ( they may give you the look, but just plead foriegn!!)
A lot of drivers in Sofia (and the rest of Bulgaria for that matter) are very aggressive, but when push comes to shove they do actually stop - Especially in Sofia - where the traffic is that bad that they can't do any great speed.
As you pointed out, there are still the mafiosa type (normally in a new bmw or Merc) that act like they own the whole city, but you get the same types wherever you're from. Don't be intimidated, most of them are just wanna be's and the if by chance you meet the real thing them the chances of them wasting their time on someone they don't know is small its not worth mentioning.

If you really want to sample some of Bulgaria, I can recommend Veliko Tarnovo, it is a much prettier city that Sofia, also Plovdiv (almost as large as Sofia but see the old town).