mberkes, I think what you have is actually a 3-speed English bike, maybe a Raleigh. It doesn't appear to be a fixed gear bike, but probably is a single speed at the moment, without its shift cable, clamps, shifter and indicator spindle. If you have access to a "bike kitchen" in your new city, you may be able to get these parts cheaply and have a more satisfactory bike. Such places also have smart, helpful people who can give you some tips about the other problems.
My favorite way to clean rusty chromed steel is a Brillo or SOS soap pad, used with water. Magic.