Given a graph property $Φ$, we consider the problem $\mathtt{EdgeSub}(Φ)$, where the input is a pair of a graph $G$ and a positive integer $k$, and the task is to decide whether $G$ contains a $k$-edge subgraph that satisfies $Φ$. Specifically, we study the parameterized complexity of $\mathtt{EdgeSub}(Φ)$ and of its counting problem $\#\mathtt{EdgeSub}(Φ)$ with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties $Φ$: the decision problem $\mathtt{EdgeSub}(Φ)$ always admits an FPT algorithm and the counting problem $\#\mathtt{EdgeSub}(Φ)$ always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property $Φ$ which, if satisfied, yields fixed-parameter tractability and otherwise $\#\mathsf{W[1]}$-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for $\#\mathtt{EdgeSub}(Φ)$ that run in time $f(k)\cdot|G|^{o(k/\log k)}$ for any computable function $f$. As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of $\#\mathtt{EdgeSub}(Φ)$. This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial $T^k_G$ of a graph $G$, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, $T^k_G(2,1)$ corresponds to the number of $k$-forests in the graph $G$. Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of $T^k_G$ at every pair of rational coordinates $(x,y)$.