Purpose of the representation theory of Lie algebra I will start with the concrete setting: Given a Lie ring $\mathcal{g}$ and a vector space $V$, which is closed under multiplication, a represenation of the Lie ring $\mathcal g$ is a structure preserving map, namely a additive and multiplicative homomorphism $\phi:\phi(a+b)=\phi(a)+\phi(b) \text{ and } \phi(a\cdot b)=[\phi(a),\phi(b)]$ where the $"[]"$ is the Lie bracket. Such a mapping is called as faithful if it is injective.
My understanding of the purpose of such representation is that we can investigate the property of the Lie ring by considering its representation, namely the elements in the vector space V.

Question: Does such representation need to be surjective? If not, I could have some elements in V, which have no correspondence in the Lie Ring. If I investigate the structure of V, I could have some larger space V than my original Lie Ring. So is it not more appropriate if we consider the image of $\phi$, which is a subspace of V. Or is it irrelevant if we are only interested in algebraic structure and not the set structure?

 A: I think the comments got a bit off-track because a vector space does not generally have a multiplication operation.  When we talk about a vector space $V$ in the context of Lie algebra representations, most people think of homomorphisms from a Lie algebra into $End(V)$, the Lie algebra of endomorphisms of $V$ (which does have a multiplication, namely function composition).  But in your case, it sounds you are talking about a $V$ which itself already has a multiplication on it.
A vector space with a multiplication is usually called an algebra.  Every associative algebra is naturally a Lie algebra with the obvious Lie bracket $[a,b] = ab-ba$. I think you are considering the embedding of an arbitrary Lie algebra into the Lie algebra of an associative algebra. An important result in this direction is the Poincare-Birkhoff-Witt theorem.
In general, the purpose of representation theory is to study an object by studying its homomorphisms into some "easier" or "standard" object that we already understand, is more concrete, is easier to compute with, etc. In your case, it sounds like $V$ is this standard object.
Depending on what we're doing, we may demand that a representation is injective or surjective, but not usually both.  If it was both injective and surjective, then that means the original object is isomorphic to a "standard" object, but usually no such standard object exists (if it did, then we probably could just study the standard object and we would not need representations).
Having said that, if you have an injective representation, then yes, it is important to determine the image of the original object inside the standard object, so that we know which part of the standard object will help us understand the original object.  But that is not the same as demanding that the representation be surjective onto the standard object. That would be demanding that the standard object be isomorphic to the original object, and is too restrictive. 
