true: "could you shut the door" would translate better to "você poderia fechar a porta?"; and "would you shut the door" to "você fecharia a porta?". they both have to do with the subjuntive.
both would sound more polite than "você pode fechar a porta?", but than the english one here "can you close the door" also have this somewhat agressive tone to it ;)
in English "can you close the door" = are you able to close the door. Of course I'm able to but will I? Correct English would be "will you or would you close the door" though "could " is often used -perhaps incorrectly - instead. However, one would use "can" if you don't know whether the person asked is able to - he or she might need a key, for example.. It is unclear to me, however, whether poder is as prescriptive as English can. Any guidance?