The running lights (low wattage filament on front signals) are tied in with the low beam power and should go off in high beam. The signal lights are separate, although the running lights are also controlled through the signal switch so that the running light on the flashing signal side turns off.
There's not much to check on the HID module. It should get 12V battery power on the input side when it should be on. If the lamp doesn't come on then, check the lamp wiring connections. Be careful and don't mess with it with the power on, there's a pretty hefty high voltage for lamp ignition that will give you quite a lift if you touch it! Some ballasts will hold a charge of this starting voltage for a long time if the lamp fails to start so be careful. A failed bulb will sometimes look fine but just not light up, sometimes they turn black or deep blue inside the bulb.