Is that part of the carpet loose or is it tacked or glued down? Most any metal edge piece or hardwood that will cover the bare spot should work.

The condensation is on the window inside of the house.

The windows appear fairly new since they are vinyl. You have warm, moist air inside our home, and cold air outside. The condensation you have is very common in the winter months and will likely get worse as the temps drop this weekend. You want to reduce the relative humidity (RH) in the home by using exhaust fans when using the shower or cooking. Lower your setting on your humidifier if you have one. Use you heat recovery ventilator if you have one. These may or may not help depending on the RH in your home. If the condensation is starting to damage the wood jamb the only other thing you can do (other then replacing the windows, and even that might not help) is to install a plastic window insulator kit.