Why is vinegar recommended for cleaning coffee pots, steam irons, and kitchen and bathroom faucets with white deposits on them? Why does it work, and what is the chemical equation for how it works?

What follows is a personal opinion and there isn't much chemical logic attached. I have seen many advertisements for books that contain 1001 uses for vinegar, 1001 uses for cabbage, 1001 uses for salt, 1001 uses for you name it. Those make interesting reading but I don't know how well they work. Vinegar is a weak acid (4-5% acetic acid) and any weak acid, such as vinegar or lemon juice (citric acid) will dissolve water marks due to carbonates. Those may be MgCO3, CaCO3, FeCO3 or Fe2(CO3)3). Those white stains one sees on the faucet handles probably are carbonates or sulfates. Calcium sulfate isn't very soluble in acid but with scrubbing and enough tries it can be done. That brown stuff on the inside of coffee pots probably is due, at least in part, to tannin and/or tannic acid. Vinegar has another property, also shared with lemon juice, of being able to complex some cations such as iron and manganese (those cause those dark brown and reddish/brown stains from some hard water supplies). That is due to the acetate ion in acetic acid or the citrate ion in lemon juice. (Salt and ice cubes work rather well for scrubbing coffee pots too.) I know I've rambled a bit but perhaps there is something somewhere in here for you to ponder.