Can I catch a particle (such as electron) and say it's left hand 
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*Why is only the left hand electron coupled to weak interaction?

*How can I tell the chirality of an electron?
 A: The chirality (left handedness/right handedness) of a fermion originates by the way it transforms under the appropriate representation of Lorentz algebra (in general the Poincare algebra). In this way, we can state that the left handed electron and the right handed electron are two fundamentally different entities (as they transform differently under the Lorentz algebra) coupled together by a Dirac mass term.
It has been observed that the left handed electron couples to a $SU(2)$ gauge field  by experiments. Since the right handed electron is a different entity altogether, there is no reason that it should also transform similarly under the same gauge field. So there is no basically reason why the right handed electron should transform similarly under weak interaction. 
The chirality of the electron can be easily told by a variety of examples. One of the simplest is colliding it with particles which also transform under weak interaction. The scattering cross sections if the electron is left handed or right handed in such a case will depend on its chirality and so you can easily tell them apart. Another way can be to determine its helicity, given you know the direction of its momentum.
