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No. Consider the case of an ellipsoid with three distinct axes, and remove the four umbilic points. Then you cannot find such vector fields on a (punctured) neighborhood of the deleted umbilics. Ha …

By the classification of surfaces, the above two cases cover all of the compact, connected surfaces $M$ with $\chi(M)\le 0$. …

In general, surfaces in $\mathbb{E}^3$ for which the principal curvatures satisfy a given functional relation $F(\kappa_1,\kappa_2)=0$ are said to be Weingarten surfaces (of type $F$), and the condition … The general theory tells you that, locally, the Weingarten surfaces of type $F$ depend on 2 'arbitrary' functions of one variable. In particular, there are lots of such surfaces locally. …

Thus, homogeneous negatively curved surfaces do not exist in $\mathbb{R}^4$. … In higher dimensions, of course, the number of types of homogeneous surfaces increases. …

I wrote an article about this problem, On surfaces with prescribed shape operator, Results in Mathematics 40 (2001), 88–121, that describes what was known about it in 2001 and gives references to the literature …

You probably will disallow this, but the following recipe does work: First, let $\nabla\phi:M\to\mathbb{R}^3$ be the (unique) vector-valued function that satisfies $$ d\phi(X) = \nabla\phi\cdot df(X) …

Any smooth compact surface smoothly embedded in $\mathbb{R}^3$ that is not the $2$-sphere must have an infinite fundamental group and hence must have infinitely many distinct (in your sense) geodesics …

To answer Joseph's questions: First, it's not impossible to integrate the geodesic flow of the hyperbolic plane in these coordinates, but the formulae I got aren't very nice, so I'm not going to type …

But the existence of an umbilic point on the sphere follows from topological considerations: The sum of the Hopf indices of the umbilics is 1 (by a theorem of Hopf) so there has to be at least one umb …

I thought about your problem and realized that there is no coordinate parametrization of the catenoid with the properties that you want. Here is my argument: First, note that, in the given $uv$-param …