Grounding helps. Also metal cases help reduce EMI.
The case acts as a simple Faraday Cage, grounded or not, the metal blocks the signals.
You don't even need metal for some frequencies, water blocks GPS frequencies, trees and leaves will do it. Water also blocks wifi.
Well noise as in sound yes, but I assume you're talking about digital noise right?
There can be RF noise, crosstalk, stray voltage...
I have seen it shut down computers and destroy nics. It's rare, but it happens.
A common point of failure in routers and modems is cheap capacitors, particularly if you live in the midwest where they have sh*tty power regulation. If a capacitor goes, there is no telling what kind of power spikes it can send down the line. It doesn't help that China stole an incomplete capacitor formula (10 years ago and are still using them), or often make substandard parts to begin with, but then you have modem and router manufacturers who under spec the power packs on these same modems and routers which only further stresses the components. Some routers are so under powered, that the power pack doesn't even supply enough power for the chipset alone, much less the rest of the circuit board or usb port. This isn't oddball unknown companies either, one very popular router was underspec'd this way when it was released, and another hugely selling (pile of garbage) from another well known manufacturer has been selling this way for years.
In many offices I work in, the owners buy cheap routers, which often are replaced every 1 to 2 years (when they become unstable). Even high end home routers only manage about 3 years under the terrible power regulation we have. I run most of my infrastructure behind a UPS (Uninterupted power supply) that contains battery backup/lightening protection/filtering/power regulation in order to make the parts last longer and run better. I found years ago that not only did parts last longer and run more stable, even in places with good regulation, but also that I could overclock higher since my power was more stable.
Getting back to this modem, it clearly has a problem, the only question is what it's currently doing, how long before it dies, and what will it do to things connected to it when it does.