I have fuller answers for you now.
By the way, in co-op, your HUD may show that the heat is off, but if your co-op partner is still outhere causing a ruckus, this will affect you. Bean and I went touring again, which inevitably leads to a cop chase or two or a dozen. We confiscated a sweet, undamaged sports car that I promptly drove up the median and got stuck. It was an elevated expressway, so when I exited I fell through the gap and landed below. We were separated at that point. I was getting tagged pretty heavily by police and the Luchadores, so I made a bee-line for the crib. I barely make it and wait for things to cool off. The HUD showed everything was clear, cops and rivals, but the second I stepped outside six cop cars and a butt-load of Luchadores came out of nowhere and took me out. Then, I respawned outside the hospital, same thing. Bean was really stirring the pot. We had to both hide in the crib to get everyone off our asses.
The cops in SR3 are amped up from the last game. They are relentless.
BTW, I took out a chase copter with one shot from my pistol (the 45 you start with).