I would drive the direction signals and brake lights directly from the corresponding switches, and just provide 5V inputs to the Arduino to let it know when the signals are active so it can display that status. The automatic headlamp would be more complicated because you will need to use an LDR or similar to detect the light level, and then operate a relay to turn the lighting on. You will need a driver circuit between the Arduino and the relay.
I don't know where you would measure the engine temperature from. If the engine has already got provision for a temperature sensor then obviously you would use that. If not, you will need to add your own, and how you do that would depend what sort of engine it is. For a water cooled engine, I would look to install an OEM coolant temp sender into the water system. For air cooled, I guess you could mechanically attach a thermister to the engine casing, although I don't know how accurately that would reflect the engine's internal temperature. In my experience engine temperature sensors are usually thermisters, which are electrically very simple to use.