Assume that we have a differential operator such as $\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$ We also then argue that if a fundamental solution has compact support, then it is supported on the origin. My follow up question is how can one then show assuming that the fundamental solution is compactly supported  that the differential operator must be of order 0?

For a constant coefficient partial differential operator P(D), the fundamental solution of P can never belong to $\epsilon'(\mathbb{R}^{n})$,i.e.have compact support. In fact,assume we have $P(D)u=f$,where u is a distribution,then u have compact support $\Leftrightarrow$ $\frac{f}{P(\xi)}$ is analytic(The result can be found in Hormander's ALPDO, volume 1,ch7.) Now,if we have $$P(D)u=\delta$$,obviously $\frac{1}{P(\xi)}$ is never an analytic function for a polynomia P.So the fundamental solution of P can not be compact supported. 


If the fundamental solution is supported at the origin, it must be a finite combination of derivatives of the Dirac distribution. This means that your original operator was of nonpositive order. 

