A few questions about the derivation of Feynman rules in specific instances have already been asked here (e.g. Sign of Feynman rules with derivative couplings, Feynman rules for coupled systems, How can we derive the Feynman rule for the ordinary QED 3-vertex?, Recipe for computing vertex factors in Feynman diagrams). However, a more general discussion seems to be missing, hence this question.
How are Feynman rules for a generic theory derived from the Lagrangian density? How are the various methodologies (e.g. second quantization, functional quantization) related to each other? When (or if) is one preferable to the others?
When and why are additional complications (like Faddeev-Popov procedures) involved in the derivation of the rules?