A polarized plug is primarily for devices that have some exposed metal attached to one side of the AC line. The common one is a lamp where the threaded side of the bulb (as opposed the the button in the middle) may be exposed. This makes sure only the grounded conductor is connected to that side.

Plug is the thing with the prongs. The thing on tend end of a extension cord is a socket. The thing on the wall that you plug it into is a receptacle. Obviously you can always use a three wire extension cord with a two wire device just as you can plug a two wire cord into a three wire (Grounded) receptacle.

There are two things that determine wire sizing in this situation. First is how much does the device draw. A two pin plug isn't going to draw more than 15A and usually much less. In that case a 14g cord is fine. The other is how much voltage drop is there over the length of the cord. Again with the small loads I doubt you'll ave a problem. If you were trying to run a refrigerator or a table saw or something at the end, that might have sen an issue.

Out door cords usually aren't worth repairing. You lose the water resistance with your repairs.

Ron,

Thanks, I'll get a new cord, probably three-prong, since they look the most common.