riemannian geometry - Geodesics on zero-curvature regions of closed surfaces of genus > 1 of non-positive curvature

Suppose that $S$ is your Riemannian surface and $X subset S$ is a flat subsurface (that is, locally isometric to $mathbb{R}$ with the usual metric). Let's suppose that $X$ has some nontrivial topology. For example, $X$ is a unit disk minus one-half of a unit disk, and the core curve of $X$ is essential in $S$.

You are correct in thinking that the universal cover of $S$ is homeomorphic to $mathbb{D}$ the unit disk. However, it is impossible to choose this homeomorphism so that the universal cover of $X$, call it $bar{X}$, embeds isometrically in $mathbb{D}$. You cannot even arrange this up to homothety. To see this, choose isometric charts for $X$ that lift to give isometric charts for $bar{X}$. After choosing where any one chart of $bar{X}$ goes in $mathbb{R}^2$ (isometrically!) the positions of all others will be determined. (This is the so-called "developing map" of $bar{X}$ and you can think of it as being similar to the process of analytic continuation of a analytic function.) The point here is that the developing map will not be injective - in fact it will have image isometric to $X$ itself.

I can't think off hand of a reasonable reference (Thurston's book is perhaps an unreasonable reference). Think about it and ask any local geometers or post more questions on MO.