If I run over pedestrians or hit other cars while driving through the city, what exactly are the consequences?
Your final rating (1-5) for each case is affected by hitting cars and pedestrians. The full list of things that affect your full score are:
Those bottom 3 are applicable to driving. If you want to get a high case rating and are a bad driver, let your partner do the driving.
If you run over pedestrians or hit other cars it lowers your total score at the end of the case. They're not the easiest to run over though as they jump out the way, but if you do you get a notification that informs you that it's not really what police officers are supposed to do and that it will impact your case grade at the end.
In terms of hitting other cars, it's exactly the same story. From memory though, you don't get the notification for it.
So in short: If you want a perfect grade for a case. Don't hit cars and don't run down pedestrians.