  Given a graph property $\Phi$, we consider the problem
$\mathtt{EdgeSub}(\Phi)$, 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 $\Phi$. Specifically, we study the parameterized
complexity of $\mathtt{EdgeSub}(\Phi)$ and of its counting problem
$\#\mathtt{EdgeSub}(\Phi)$ with respect to both approximate and exact counting.
We obtain a complete picture for minor-closed properties $\Phi$: the decision
problem $\mathtt{EdgeSub}(\Phi)$ always admits an FPT algorithm and the
counting problem $\#\mathtt{EdgeSub}(\Phi)$ always admits an FPTRAS. For exact
counting, we present an exhaustive and explicit criterion on the property
$\Phi$ 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 $\#\mathtt{EdgeSub}(\Phi)$ 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 $\#\mathtt{EdgeSub}(\Phi)$. 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 $(x,y)$.
