A "fixie" is, generally a fixed gear bicycle. There is no freewheel. When the pedals move, the rear wheel moves. Pedal backwards, the rear wheel moves backwards. Stop pedalling, the rear wheel stops.
Full restoration means different things to different people, I suppose. I personally think that a vintage bicycle should be restored with original spec parts. Otherwise, I'd call the bike "rebuilt."
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Religion is a good thing for good people and a bad thing for bad people. --H. Richard Niebuhr