No, you have it backward. When they go bright, the alternator is working. When they go dim, it isn't. Batteries are just storage devices that store electricity. The alternator produces the electricity that your vehicle uses and provides enough extra to store some in the battery for next time you need to start the car. Get a volt meter, they're cheap to buy and worth the investment. Check the battery with the key off. It should be a little over 12 volts but certainly not much below 12 volts. Start the engine, either using a jump start or by re-charging the battery with a charger. After the engine is started, remove all outside sources of electricity (jumper cables or charger) and check the battery voltage again. It should be somewhere around 13.5 volts and certainly over 12 volts. You should be able to turn on headlights, heater blower and just about anything else and the voltage shouldn't drop below 13.5 volts. You may need to bring the engine RPM's up to about 1000 or 1500 (which isn't much if you don't have a tach) to maintain 13.5 volts. If the alternator can't give you 13.5 volts, you need a new one.
A bad battery or faulty battery connection.
The new alternator is bad happens all the time.
Battery drain would depend on what else is on, for example headlights, courtesy lights, glove box light etcetra. If there is nothing electrical on then there should be no drain.
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.
leaving your lights on for a long period of time.. headlights or just playing the radio and leaving it play for a long time
Yes, if the relay was stuck it would drain the battery.
Have your Alternator tested. Sounds like the entire engine is running off just the battery.
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
Switch? both headlights?
Headlights have two filaments, one for dim, one for bright. From your description it appears that one of the high beam filaments is burned out. Replace the bulb.
Because if you reversed the polarity at the battery it would blow the fuse through which current flows to the drain.