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The opto-isolators make everything pretty-safe for your car and the Arduino.
Your Arduino inputs from the opto-isolators are connected to ground... That won't work. 
They should be connected to the collector (where the resistor connects to the opto-isolator).
If you enable the optional internal pull-up resistor (inside the ATmega chip) you can use it instead of the 4.7K you're using on the Arduino-side of the isolator. [u]This Example[/u] shows how to use the internal pull-up with a pushbutton switch. The opto-isolator can be used the same way.
You don't need the reverse diode across the LED in the opto-isolator.
The 4.7K resistor on the LED side of the isolator is a little high in value. 1K is more like it. The datasheet will give you the current range and voltage for the LED, but if we assume 2V across the LED and 10V across the resistor, we can calculate the current as 10V/1K = 10mA, and that's probably about right.
Make sure you've got enough current available from 5V power supply for the LED strips. "Worst case" is about 60mA per LED (white at full-brightness). That can add-up fast! And it's best if the LED strip has it's own power connection so the current doesn't have to flow through the Arduino board.
Regular opto-isolators will invert the logic (high becomes low) and sometimes a light in a car is turned-in by making the ground connection so you might have to reverse your logic in software. So, try something like that pull-up example to check what you're reading when you hit the breaks, etc.
P.S
[u]This kind[/u] of wire splice allows you to tap-into your car's wiring without cutting the wires.
