A lighter socket for a TV or other general purpose use would work just fine for a phone charger or other small device powered by an inverter (provided, of course, it has power supplied to it, as apparently the original poster's does not). I would suggest a limit of about a hundred watts, less if you find 12V fuses for the circuit are blowing.
Chris is correct about the black box doohicky on the right. This is legally required in practically every state if the television is in view of the driver, and at least theoretically in several states whenever the TV screen is in front of the driver's seat headrest, regardless of whether it's in view of the driver.