Hi, You use to be able to buy a rubber (Blade) and/or a thin Bristle long brush that fitted into a slot either cut in the bottom of the door or set into a board that was attached to the back of the door. Over the years many doors had the bottom shaved off to allow the door to pass over carpet or linoleum. Once these are removed to achieve the polished floor look the caps left are huge and need to be addressed

There are several kinds of weatherstripping that can be added to an existing door. Some of them almost invisible, others easier to install. And it sounds like you need to insulate under your floor! Or at least put down a throw rug.