x=x(u), z=u, how did you see x=z?
To prove that meridians of a revolving surface are geodesics, notice that the surface is symmetric about the generatrix( the meridian). To be more precise, a reflection about the plane spanned by the z-axis and a generatrix, keeps the surface unchanged. Pick two nearby points p, q at the generatrix, the generatrix must be the shortest path between these two points, otherwise suppose there is another curve c connecting p, q, the reflection c' has the same length of c, so c' is shortest too. This contradicts the assertion that geodesics are locally unique.