Regarding your initial problem;
Periodic ticking sound in GNOME that doesn't stop unless I mute.
Does the sound only occur when you move your mouse? I've had a few Dells (even laptops from Dell) do that. Usually it's been with a wired mouse (USB or PS/2). Oddly enough, it also happened when a AM/FM radio was on nearby.
As for your dead battery; if you're in the habit of keeping the system plugged in, just remove the battery.
Again, I have two laptops that now live as servers, and their batteries went kaput years ago, so out they went.