Well depending on the size of the power inverter, it might not be a good idea to hardwire it to the cigarette lighter because the cig lighter wire is extremely thin. If you can, find a thicker gauge wire to hardwire the power inverter to, and install a switch somewhere, and yes it would be a good idea to wire an in-line fuse with the power wire as well...

There are many different sources that you can wire the inverter to that would turn off when the car is off, it doesn't necessarily have to go to the cigarette lighter.