Yes, they can stay plugged in when not in use. However even if the car is not running, the charger will continue to use a small amount of battery power, even if there is no device plugged in. This could affect the battery of your car if it is not driven for long periods of time.
No, there should be no interference. If there is an issue, try to place the backing at a different location on the phone. Note: If you use wireless charging (for compatible phones), the metal plate might interfere with charging. If so, remove the metal plate before charging.