I would say no! You have other issues with your head lights, a burnt connector or loose ground wire, cheap bulbs or loose fuse box.
Did you change the multi switch?

 

I didn’t replace the multi-function switch since there is no Auto Light On issue now. When I covered the Light Sensor with paper, I found Auto Light On works normally. However, Auto Light On didn’t work on rainy days before. That is what I don’t understand.

 

Battery voltage when the engine is on is actually the alternator output, that varies by load. If you want to check the battery do it engine off.

 

Right. Now I think the Voltage Regular of the Alternator might be malfunctioning, causing the voltage to be too high and thus draining the battery. Does this make sense ?

 

Voltage from the alternator charges the battery, it doesn’t drain the battery.
check the static voltage of the battery, engine not running, ignition key off and out of the ignition switch. That should be 12.4 V or higher

 

The regulator is the BCM. The demand comes from that strange thing wrapped around the negative battery cable.
I am sensing that you are expecting things of the auto lights that are not part of it. In AUTO mode the low beams are on always no tail lights or marker lights WHEN the trans is not in park and the key is on: When it gets dark the tail lights and marker lights go on. When the key is turned off AND the driver door open then closes the head lights and tail lights go off on a timer In ON mode the head lights and tail light are on until you turn them OFF

 

You're right. A month ago, I checked the battery after turning off the engine, and the voltage was 12.1 volts. Two weeks ago, I checked it again, and the voltage was 12.4 volts. I often hear clicking noise when I start my car in Winter.

 

12.4 volts is a dead battery. should be above 12.7.

 

Yes, a good battery should hold at least 12.4 volts for quite some time, but these cars seem to prefer a little more than that. I would definitely just replace a battery that won't hold 12.6

First step is to check the voltage at the battery posts after the car is set overnight before you touch anything else.

Lead acid batteries need to remain charged. A battery that has sat in the cold for any length of time in a discharge state can have internal shorts due to sulfating. This can happen even with a very new battery. A battery that self discharges to 10.5-10.8 V probably has such a short in one of its six cells.

A newish battery might be repairable with a battery conditioner, but may not be worth the trouble.

 

Yes. I'm planning to buy a car battery. Do you know where I can find one at a good price? Including online shopping. Thanks.