No. The power to illuminate the Daytime Running Lights (DRL's) is supplied by the alternator, and does not have to be supplied by the battery. Once a vehicle is started and the engine is running you could remove the battery from the car entirely and everything requiring current would still operate.
Yes, headlights drain battery power, but it is only major if you leave them on when you turn the car off, because the car might not start up after a bit.
If the alternator is installed incorrectly, yes, it can drain the battery.
Headlights, And Any other application you have running (ex. Radio).
If the car is on (Ignition is on, engine is running) then no. Nothing should drain the battery when the car is running, there is a device called an alternator with a belt that usually goes around the crankshaft that transforms the mechanical energy into electrical energy. As long as you have the car running, the battery should not die. If so, you have a mechanical problem.
It might
Headlights run off the battery not the ignition...
If the engine is running, then no. The alternator constantly keeps the battery charged.
If the donor vehicle is running and has a properly functioning alternator, no.
If your electrical system is working as it should, using your headlights will not drain the battery. Of course that is assuming your engine is running. More than likely your alternator is not putting out enough voltage to keep the battery charged with the headlights on. Take your car to a trusted mechanic or auto parts store and have your charging system checked. You may need a new alternator. A loose drive belt can also cause this so make sure the belt is in good working order and adjusted correctly.
The Legacy subwoofer will really drain your battery. It's not a good idea to run it while you're not running the engine.
mainly leaving your lights on when the engine is of or leaving the keys in the ignition
While the car is running the alternator should be recharging the battery faster than stereo can drain it. I suspect you either have a problem with the alternator or its time to get your battery replaced. Until you get it fixed, if you don't turn on the radio, it wont drain your battery :) If it is a high power model, it needs to be wired directly to the battery.