Some questions on closed geodesics I can't imagine the following claim.

Suppose we have a surface and a closed geodesic on it. If you try to keep the starting point and the initial direction of a closed geodesic
but slightly deform the surface, it might happen that after the deformation the
geodesic is not even closed any more!

Q1: How is that possible? any example?

A theorem about number of closed geodesics state that

Theorem (Grove–Gromoll): For any metric on the 2–dimensional
sphere with all geodesics closed, the geodesics have all the same length.

Q2: What about Ellipsoid? Geodesics of Ellipsoid have all the same length?

Q3: It seems that existence infinitely many closed geodesics on sphere is an open problem. (Yes?) Does it this means that diffeomorphisms may not preserve geodesics? Isn't it strange? any example?
 A: I will just address Q3. The existence of closed geodesic on 2 sphere is a very classical problem.
In 1917, Birkholf showed that there is always one closed geodesic on any Riemannian 2-sphere. He used the min-max argument. The proof can be found in here, here for example.
Then there is the Theorem of three geodesics: Quoting from Wikipedia: In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics, and in 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the conjecture, which was later found to be flawed. The proof was repaired by Hans Werner Ballmann in 1978.
Going back to your question, the question is settled at around 1992, when J. Frank proves the existence of infinitely many closed geodesics on a Riemannian 2-sphere with positive Gauss curvature (here). They used ideas from dynamical systems, which goes back to Birkholf. The assumptions on Gauss curvature is later dropped by V. Bengert. So it is no longer an open problem.
A: To start, note that the geodesics of the round sphere $S^2$ are exactly the great circles. All great circles are a) closed b) of equal length. This is should be proved in any text on differential or Riemannian geometry.
Q1) The sphere is diffeomorphic to any ellipsoid (see Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.) which can have infinitely many geodesics which are not closed. Check out the images on this Wikipedia page.
Q2) The Wikipedia link in Q1) gives examples of geodesics on an ellipsoid which are not closed, and therefore the theorem of Grove and Gromoll does not apply here. As Ted says in the comments, it is trivial to find geodesics on an ellipsoid with different lengths (can just take the ones along the principal axes).
Q3) I do not know if this is an open problem, but it is not a consequence of Q1) and Q2).
Yes this means diffeomorphisms may not preserve geodesics, as in Q1).
No this is not strange; as Ted suggests in the comments, diffeomorphisms in general have no obligation to preserve Riemannian structure.
