Sound engineers deal more with the recording and electronics, you want an acoustician. That is a big hobby of mine and I've helped design a few listening rooms and home theaters.

That being said, those foam pads on the wall aren't a meme, they just don't do what people think. They absorb reflections of sound, meaning they make things quieter in the room that made noise in that room. If you've ever clapped your hands in a gym and heard the echo bounce around, that's what it stops.

What they don't do is stop sound from traveling through a wall when it's hit. That does indeed require either density or distance (an air gap). Thickness is usually just an easy way of increasing density so I don't actually count it separately.

If you can explain your situation a little more (house vs apartment, what sound is bugging you, etc) I can maybe suggest more. Sometimes it can be sound passing through unexpected holes (around windows is an exceptionally notorious spot, some installers spray expanding foam now but not all) and not actually transmitting through the wall itself.

It depends what kind of sound.

In general though, you heard right. There is a product called "mass-loaded vinyl" that I used (by soundsulate, I think) between TV and my wife's study - but if you have access to the wall cavity, rockwool is very good

If you don’t own the place, forget about it. If you do, and you’re serious, put up a metal channel on the wall to create space and use sound damping material in between the wall and channel, then put another layer of drywall with sound damping material. Low frequencies make vibrations through anything connected so even the ceiling could be a resonator. You’re gonna have to either isolate the room with the sound or the room you want to be sound proof. It’s a big job.