That's silver tarnish, your keys are tarnishing because hard rubber (the material your mouthpiece is made of) causes silver to tarnish. This is why manufacturers often recommend storing the mouthpiece outside of the case or putting it in its own little bag or pouch.
If you want to polish it, get a Yamaha medium silver polishing cloth.
https://www.amazon.com/Yamaha-YAC-1110P2-Medium-Polishing/dp/B01M6C93KN
Unfortunately now that your silver is no longer "bright" your sound will go too "dark" and will sound dead...
(Kidding of course, it's just cosmetic)
For keeping silver keys shiny it's a good idea to use a polishing cloth once a month. My favorite is the Yamaha:
https://www.amazon.com/Yamaha-YAC-1110P2-Medium-Polishing/dp/B01M6C93KN
You also want to wipe the keys regularly with a microfiber cloth after you handle the instrument. You can get these at an automotive store or if you want a nice one get an eyeglass wipe.
I've never heard of wood getting foggy, you can try taking a cloth with a drop (emphasis on a drop, a little goes a long way) of almond oil or bore oil and wiping the wood. Be sure to avoid any pads.
I'm new to meckb but as a flutist, i can recommend you to get a cleaning cloth for the stainless steel parts that are dirty, these cloth wont scratch the surface when cleaning.
You can also buy a polishing cloth but every time you polish, a small layer is polished off, so it might affect the sound of the kb (not sure how thick is your steel parts, but on a flute it makes a whole lotta diff). Hope that helps!
It tarnish and can be removed with a polishing cloth:
https://www.amazon.com/Yamaha-YAC-1110P2-Medium-Polishing/dp/B01M6C93KN
But also don't leave your case open, otherwise it will attract dust. Make sure you vacuum out the case regularly regardless too.
grab a polishing cloth and go to town before you do anything heavier