classical mechanics - Generalized definitions of Lagrangian and Hamiltonian functions

When we enter into the scope of Analytical mechanics we usually start with these two primary notions: Lagrangian function & Hamiltonian function

And usually textbooks define Lagrangian as $L=T-V$ and Hamiltonian as $H=T+V$ where $T$ is Kinetic energy and $V$ is Potential energy. But as we proceed it turns out Lagrangian and Hamiltonian may not always have these values and this occurs in just some special cases and there is a more general definition.

My question is:

1. 
   

   What is the generalized definitions of these two functions?

   
   


2. 
   

   Is there a general defining equation where the mentioned values could be extracted as a special case?

   
   

EDIT: A third question

Do we know about the notions of "Kinetic energy" and "Potential energy" beforehand or are they defined after the Lagrangian gets its famous form $L=T-V$ and then we assign $T$ and $V$ their respective definitions?

EDIT 2: A fourth question

What if there were non-conservative fields? Then obviously $V$ term couldn't show their effects(since Potentials are always associated with conservative fields). Then how could we bring the effects of non-conservative fields into play?