There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). 

In the case of geodesics representing some non trivial homotopy class of closed curves on a (not simply connected) manifold, I understand the importance/usefullness of knowing that such a homotopy class (element of the fundamental group) can be represented by a geodesic. But what about simply connected manifolds? In dimension 2 the simply connected surfaces are $S^2, \mathbb{R}^2$ and $\mathbb{H}^2$; according to Lusternik-Fet, $S^2$, being compact, admits non trivial closed geodesics (whereas the other two do not), but they are all homotopically trivial; I don't get what these closed geodesic tell about the topology of $S^2$; my feeling is that maybe the whole situation is quite trivial in dimension 2, but I'm not much familiar with manifolds of dimension 3 or higher (spheres, euclidean and projective spaces apart)...so any interesting example is welcome!

I know that existence results are always of fundamental importance on their own, but here I can't figure out which are the implications of the existence of closed geodesic (especially in the simply connected case, as pointed out above).

To sum up, I'm interested in the following questions:

1) what if a manifold has a closed geodesic?

2) what if a manifold has no closed geodesic? (can I say that then the manifold is simply connected?...the punctured euclidean plane seems a counterexample...what if I restrict my question to complete manifolds?)

3) what if a manifold has more than one closed geodesic? or even infinite? (here I mean "distinct" geodesics, in some sense)

4) what if every geodesic is closed?

I would appreciate explicit examples as well as theorems or references on the subject.