How to find cubic non-snarks where the $\min(f_k)>6$ on surfaces with $\chi<0$?
For orientable surfaces, here's a representative element of a family of non-snarky cubic graphs on an $n$-torus with $4n-2$ vertices, $6n-3$ edges and a single $(12n-6)$-sided face.
If it is a problem that some pairs of vertices have more than one edge going between them, that can easily be fixed with some local rearrangements (which can be chosen to preserve the green-blue Hamiltonian cycle and thus non-snarkiness).
For non-orientable surfaces, I think it is easiest to start by appealing to the classification theorem and construe the surface as a sphere with $k\ge 2$ cross-caps on it. Now if you start with a planar graph and place a cross-cap straddling one of its edges (so following the edge from one end to another will make you arrive in the opposite orientation), the net effect is to fuse the two faces the edge used to separate.
Therefore, start with an arbitrary planar non-snarky cubic graph with $2k-2$ vertices and $3k-3$ edges. Select an arbitrary spanning tree (comprising $2k-3$ edges), and place cross-caps on the $k$ edges not in the spanning tree. This will fuse all of the original graph's faces, and we're left with a graph with a single $(6k-6)$-sided face.
In each of the above cases, if you want more faces, simply subdivide the single one you've already got with new edges. The face itself, before you glue its edges together to form the final closed surface, is just a plain old $(12n-6)$- or $(6k-6)$-gon, so you can design the subdivision as a plane drawing.