Does the shape of an orbit determine its stability?
Specifically, is an elliptical orbit stable?
Well, remember the main concept of stability: "start close, stay close" (in the sense of Lyapunov) + "convergence" (asymptotically). So, if your dynamical system has a closed (isolated) trajectory to which all nearby trajectories converge, then it is stable (at least locally). This is irrespective of its shape. Conversely, if any nearby trajectory diverges away from it, then it is unstable (or half-stable).
Also, is a converging elliptical orbit asymptotically stable?
Remember that the concept of stability is related to an equilibrium point, or a limit cycle (a closed trajectory as explained above). So, the point/orbit towards which the elliptical orbit in question converges, is what can be considered in terms of stability.