About connecting the leds, please use Google. Pages and pages full about it.
12V is fine for the Vin of an Arduino if it doesn't need to provide real power (which it doesn't seem to do here) But the 12V from a vehicle isn't really 12V. When it's running it's getting close to 15V. It can also be very noisy. So I would get a DCDC buck converter, set it to around 8V, add a +-220uf and a 100nF cap to that and connect it to the Vin of the Nano.
Or the real easy solution. Get a (decent) USB car charger, plug it in and connect the USB of the Arduino to it.