Are you allowed to question the police about their service or conduct?

Yes, of course. Assuming it doesn't interrupt an ongoing investigation. But they probably won't take kindly to it, They do a dangerous job and don't like being questioned about how they are doing it. If you see a police officer do something way out of line, call the department and talk to a supervisor. Asking a police officer for his badge number alos isn't the smartest move. It tends to make them angry. But if the officer has done something seriously out of line, ask for a badge number, or their full name. Or if you live in a city where police officers have designated units, get their unit number. And call a supervisor.