In lectures on the standard model I recently saw, in writing down the SM Lagrangian the professor was careful to refer to fields like $e^c$ as a conjugate electron rather than an antielectron. He also mentioned that all the fields he wrote down were left-handed Weyl spinors.
First Question: What is the relationship between conjugate electrons and antielectrons?
Second Question: Are fields in the SM Lagrangian left-handed spinors by convention, or is this physically meaningful?
Third Question: Is there any connection between charge conjugation and handedness of a spinor?