Thanks for playing and for the feedback!
You do take damage from hitting cars, though not a lot. If I remember the game's stats right, getting hit takes about 3% of your base health. It's not much but you can easily get hit by a ton of cars, especially in the last level, so it adds up pretty fast. I also have a bad habit of making jam games too hard, so this time I made a deliberate effort to keep things relatively easy (I may have over corrected though).
I'm not sure what you're asking about in your second point. Do you mean that police car health should be increased, or displayed? Thanks.