Is classifier able to say there's no-such-case?
Taking your questions one after the other:


  Assume that I have a classifier which can classify left hand / right
  hand well. I am curious whether it can decide whether there's a hand
  in the image?


No. It cannot. A model which is trained on hand/no-hand can do that. Not this. At most, it shall predict one of the either (left/hand) with a very low confidence score. But, it definitely is not a nice idea to use it for a hand/no-hand identification.


  But there's so many things that are "no-hand" such like dogs, cats,
  flies... I don't think it is possible to collect all those possible
  "no-hand" data


You are trying to do build a binary hand.no-hand classification model, not a hand/cat/dog/... style multi-label classification model. So, you just need to have a bunch of images with a hand in it, and some more which doesn't have one.


  Would it be reasonable that we use some activation functions like
  [sigmoid] as the final outputs for the classifier(neural network) and
  we consider the outputs for both left/right classes as the scores that
  classifier think how far the input is from the left/right hands
  classes.


Possible, but not advisable.