The Norm function assumes by default that your vector is complex and that the derivatives are in the holomorphic sense. Your expression will contain Abs, and the absolute value is not differentiable in the complex sense. I wish that Norm had an option to assume that the vectors are real. In the meanwhile you can work around the problem by removing the Abs:

D[Norm[{x, y}], x]
D[Norm[{x, y}] /. Abs -> Identity, x]
D[Norm[{x, y}], x] /. Abs -> Identity

Look at the FullForm of your expression before you try to take the derivative and think about what taking the derivative of each component of that might be.

I suspect that Norm is the source of some of your problems.

Abs' is the derivative of Abs and cannot be simplified further by Mathematica unless you can provide sufficient assumptions that Mathematica can determine the argument will always be positive or will always be negative.