So this weird buzz-electrical sound is coming even when the car's not started but accessories are on.
It's coming behind the glove box, I think. Where should I look for this noise to stop?

 

Sounds like a bad relay.

Remove the glove box to check.

 

Sound is now gone, after parking for 30 minutes. How serious is bad relay gonna affect driving? Should I repair this in shops or is this easy job for newbie DIY'er?

https://photos.app.goo.gl/EnzhRBkLrY83bZRR8
I took this video thinking it was coming from fan.

 

A relay is easy enough to replace , pop it out put one in . Are you replacing one needlessly ? If it's making noise or hot to the touch I'd replace it or even carry one on board , You can swap a similar one for another to see if it is the problem .

 

I just pulled couple of relays from pick n pull (didn't knew I had to release the tab while removing.. so I broke 2 relays making them useless. Sorry world)

So full context for my car: DRL light is on until I turn low beam on (Canadian LX) but not actually functioning, switching to high beam from low beam also doesn't respond. No dash indicator.

Searching a bit more with this symptom led me to "DRL module solder" but since I didn't had soldering kit (and I was going to pick n pull anyways) I just pulled one out.

Could be the noise from DRL module's fault?

 

Yes it can. When the DRL module goes bad they will buzz. Same with relays.
The DRL system uses the high beam circuit on low voltage.
Also after-market stereos can cause issues to various electrical components when you hook up the illumination harness of the headunit. Don't connect the illumination connection.

 

Okay, I'll try to replace DRL module. Should I disconnect battery whilst doing this?

The car does have after-market radio/music player. Will that have any affect?

 

Yes, it can. The illumination connection is not to be used when installing an after-market stereo in these cars. But I think you just have a bad relay on its way out behind the glovebox.