This may be more a post for more information about the high power 1/3/5W LEDs, but here's what I want to do:
I noticed my inside car lights are, for one, pretty dim, and when taking off the cover it shows that they are 8W incandescent lamps. Would replacing these lamps with a high power LED work out well? I think my main concern is that I'm scared that they will get hot and cause damage, do they? Are there any other foreseeable problems that this modification might cause?
Generally you'll be fine. In fact, the LEDs will put out far less heat then your existing bulbs. Just make sure to use the appropriate series resistor.
One thing to watch for is that sometimes the filament of the bulb is used as a resistor in the light dimmer circuit. If you replace with LEDs you may find that the courtesy function no longer works.
How much resistance would a typical bulb have? You'll probably have to include a current-limiting resistor in the circuit anyway to keep the LEDs from drawing too much.